new proofs of bapat and sivasubramanian s theorems abstract recently bapat and sivasubramanian 1 presented formulas for the determinant and the inverse of the product distance matrix of a tree furthermore they defined a bivariant ihara selberg zeta function of a graph and gave its determinant expression we present new proofs for three results of bapat and sivasubramanian

generalized characteristic polynomials of graph bundles abstract in this paper we find computational formulae for generalized characteristic polynomials of graph bundles we show that the number of spanning trees in a graph is the partial derivative at 0 1 of the generalized characteristic polynomial of the graph since the reciprocal of the bartholdi zeta function of a graph can be derived from the generalized characteristic polynomial of a graph consequently the bartholdi zeta function of a graph bundle can be computed by using our computational formulae

partial words and a theorem of fine and wilf a partial word is a word that is a partial mapping into an alphabet we prove a variant of fine and wilf s theorem for partial words and give extensions of some general combinatorial properties of words

some properties of graphs determined by edge zeta functions in 1989 hashimoto introduced an edge zeta function of a finite graph which is a generalization of the ihara zeta function the edge zeta function is the reciprocal of a polynomial in twice as many indeterminants as edges in the graph and can be computed via a determinant expression we look at graph properties which we can determine using the edge zeta function in particular the edge zeta function is enough to deduce the clique number the number of hamiltonian cycles and whether a graph is perfect or chordal finally we present a new example illustrating that the ihara zeta function cannot necessarily do the same

a new bartholdi zeta function of a digraph we introduce a new type of the bartholdi zeta function of a digraph d and give a determinant expression of it furthermore we define a new type of the bartholdi l function of d and give a determinant expression of it as a corollary we obtain a decomposition formula for a new type of the bartholdi zeta function of a group covering of d by new bartholdi l functions of d

deriving good ldpc convolutional codes from ldpc block codes low density parity check ldpc convolutional codes are capable of achieving excellent performance with low encoding and decoding complexity in this paper we discuss several graph cover based methods for deriving families of time invariant and time varying ldpc convolutional codes from ldpc block codes and show how earlier proposed ldpc convolutional code constructions can be presented within this framework some of the constructed convolutional codes significantly outperform the underlying ldpc block codes we investigate some possible reasons for this convolutional gain and we also discuss the mostly moderate decoder cost increase that is incurred by going from ldpc block to ldpc convolutional codes

functionals of brownian motion localization and metric graphs we review several results related to the problem of a quantum particle in a random environment in an introductory part we recall how several functionals of brownian motion arise in the study of electronic transport in weakly disordered metals weak localization two aspects of the physics of the one dimensional strong localization are reviewed some properties of the scattering by a random potential time delay distribution and a study of the spectrum of a random potential on a bounded domain the extreme value statistics of the eigenvalues then we mention several results concerning the diffusion on graphs and more generally the spectral properties of the schrodinger operator on graphs the interest of spectral determinants as generating functions characterizing the diffusion on graphs is illustrated finally we consider a two dimensional model of a charged particle coupled to the random magnetic field due to magnetic vortices we recall the connection between spectral properties of this model and winding functionals of planar brownian motion

small linearly equivalent g sets and a construction of beaulieu abstract two g sets g a finite group are called linearly equivalent over a commutative ring k if the permutation representations k x and k y are isomorphic as modules over the group algebra kg pairs of linearly equivalent non isomorphic g sets have applications in number theory and geometry we characterize the groups g for which such pairs exist for any field and give a simple construction of these pairs if k is q these are precisely the non cyclic groups for any non cyclic group we prove that there exist g sets which are non isomorphic and linearly equivalent over q of cardinality 3 g 2 also we investigate a construction of p beaulieu which allows us to construct pairs of transitive linearly equivalent s n sets from arbitrary g sets for an arbitrary group g we show that this construction works over all fields and use it construct for each finite set p of primes s n sets linearly equivalent over a field k if and only if the characteristic of k lies in p

characterizations of pseudo codewords of low density parity check codes an important property of low density parity check codes is the existence of highly efficient algorithms for their decoding many of the most efficient recent graph based algorithms e g message passing iterative decoding and linear programming decoding crucially depend on the efficient representation of a code in a graphical model in order to understand the performance of these algorithms we argue for the characterization of codes in terms of a so called fundamental cone in euclidean space this cone depends upon a given parity check matrix of a code rather than on the code itself we give a number of properties of this fundamental cone derived from its connection to unramified covers of the graphical models on which the decoding algorithms operate for the class of cycle codes these developments naturally lead to a characterization of the fundamental cone as the newton polyhedron of the hashimoto edge zeta function of the underlying graph

inverses of q distance matrices of a tree the determinant and the inverse of the distance matrix of a tree have been investigated in the literature following the classical formulas due to graham and pollak for the determinant and due to graham and lovasz for the inverse we consider two q analogs of the distance matrix of a tree and obtain formulas for the inverses of the two distance matrices yan and yeh have previously obtained expressions for the determinants of the two distance matrices some related results are proved

a new weighted ihara zeta function for a graph abstract we define a new weighted ihara zeta function of a graph g and present its determinant expression we present a decomposition formula for the new weighted ihara zeta function of a regular covering of g furthermore we introduce a new weighted ihara l function of g and give determinant expressions of it as a corollary we present a decomposition formula for the new weighted ihara zeta function of a regular covering of g by its new weighted ihara l functions as applications we give new proofs for the results of kempton on the spectrum of the transition probability matrices for non backtracking random walks for regular graphs and semiregular bipartite graphs

the coefficients of the reduced bartholdi zeta function abstract in this paper we establish a new zeta function based on the bartholdi zeta function for an undirected graph g called the reduced bartholdi zeta function we study the relation between its coefficients and the structure of the graph and demonstrate that the coefficients count the star subgraphs in the symmetric digraph d g moreover we investigate the properties of semi principle minors extracted from the adjacency matrix of the oriented line graph of g we also present a general formula for calculating all the coefficients of the reduced bartholdi zeta function

riemann roch and abel jacobi theory on a finite graph abstract it is well known that a finite graph can be viewed in many respects as a discrete analogue of a riemann surface in this paper we pursue this analogy further in the context of linear equivalence of divisors in particular we formulate and prove a graph theoretic analogue of the classical riemann roch theorem we also prove several results analogous to classical facts about riemann surfaces concerning the abel jacobi map from a graph to its jacobian as an application of our results we characterize the existence or non existence of a winning strategy for a certain chip firing game played on the vertices of a graph

efficient computation of ihara coefficients using the bell polynomial recursion abstract the ihara zeta function has proved to be a powerful tool in the analysis of graph structures it is determined by the prime cycles of a finite graph g v e and can be characterized in terms of a quasi characteristic polynomial of the adjacency matrix t of the oriented line graph associated to g the coefficients of this polynomial referred to as ihara coefficients have been used to characterize graphs in a permutation invariant manner and allow for an efficient evaluation of the ihara zeta function in this paper we present a novel method for computing the ihara coefficients we first establish a characterization of the ihara coefficients in terms of complete bell polynomials and by exploiting a recursive relation for the latter we show how the ihara coefficients can be efficiently computed in o e 2 provided that the eigenvalues of t are known

zeta functions of quantum graphs in this paper we construct zeta functions of quantum graphs using a contour integral technique based on the argument principle we start by considering the special case of the star graph with neumann matching conditions at the center of the star we then extend the technique to allow any matching conditions at the center for which the laplace operator is self adjoint and finally obtain an expression for the zeta function of any graph with general vertex matching conditions in the process it is convenient to work with new forms for the secular equation of a quantum graph that extend the well known secular equation of the neumann star graph in the second half of this paper we apply the zeta function to obtain new results for the spectral determinant vacuum energy and heat kernel coefficients of quantum graphs these have all been topics of current research in their own right and in each case this unified approach significantly expands results in the literature

expander graphs and their applications a major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists since expander graphs the protagonists of our story come up in numerous and often surprising contexts in both fields but perhaps we should start with a few words about graphs in general they are of course one of the prime objects of study in discrete mathematics however graphs are among the most ubiquitous models of both natural and human made structures in the natural and social sciences they model relations among species societies companies etc in computer science they represent networks of communication data organization computational devices as well as the flow of computation and more in mathematics cayley graphs are useful in group theory graphs carry a natural metric and are therefore useful in geometry and though they are just one dimensional complexes they are useful in certain parts of topology e g knot theory in statistical physics graphs can represent local connections between interacting parts of a system as well as the dynamics of a physical process on such systems the study of these models calls then for the comprehension of the significant structural properties of the relevant graphs but are there nontrivial structural properties which are universally important expansion of a graph requires that it is simultaneously sparse and highly connected expander graphs were first defined by bassalygo and pinsker and their existence first proved by pinsker in the early 70s the property of being an expander seems significant in many of these mathematical computational and physical contexts it is not surprising that expanders are useful in the design and analysis of communication networks what is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness in mathematics we will encounter e g their role in the study of metric embeddings and in particular in work around the baum connes conjecture expansion is closely related to the convergence rates of markov chains and so they play a key role in the study of monte carlo algorithms in statistical mechanics and in a host of practical computational applications the list of such interesting and fruitful connections goes on and on with so many applications we will not even

identities for minors of the laplacian resistance and distance matrices it is shown that if l and d are the laplacian and the distance matrix of a tree respectively then any minor of the laplacian equals the sum of the cofactors of the complementary submatrix of d up to sign and a power of 2 an analogous more general result is proved for the laplacian and the resistance matrix of any graph a similar identity is proved for graphs in which each block is a complete graph on r vertices and for q analogues of such matrices of a tree our main tool is an identity for the minors of a matrix and its inverse

ihara s zeta function for periodic graphs and its approximation in the amenable case abstract in this paper we give a more direct proof of the results by clair and mokhtari sharghi b clair s mokhtari sharghi zeta functions of discrete groups acting on trees j algebra 237 2001 591 620 on the zeta functions of periodic graphs in particular using appropriate operator algebraic techniques we establish a determinant formula in this context and examine its consequences for the ihara zeta function moreover we answer in the affirmative one of the questions raised in r i grigorchuk a uk the ihara zeta function of infinite graphs the kns spectral measure and integrable maps in v a kaimanovich et al eds proc workshop random walks and geometry vienna 2001 de gruyter berlin 2004 pp 141 180 by grigorchuk and uk accordingly we show that the zeta function of a periodic graph with an amenable group action is the limit of the zeta functions of a suitable sequence of finite subgraphs

quantum chaos on discrete graphs adapting a method developed for the study of quantum chaos on quantum metric graphs kottos and smilansky 1997 phys rev lett 79 4794 kottos and smilansky 1999 ann phys ny 274 76 spectral functions and trace formulae for discrete laplacians on graphs are derived this is achieved by expressing the spectral secular equation in terms of the periodic orbits of the graph and obtaining functions which belong to the class of functions proposed originally by ihara 1966 j mat soc japan 18 219 and expanded by subsequent authors stark and terras 1996 adv math 121 124 kotani and sunada 2000 j math sci univ tokyo 7 7 finally a model of classical dynamics on the discrete graph is proposed it is analogous to the corresponding classical dynamics derived for quantum graphs kottos and smilansky 1997 phys rev lett 79 4794 kottos and smilansky 1999 ann phys ny 274 76

bartholdi zeta and l functions of weighted digraphs their coverings and products since a zeta function of a regular graph was introduced by ihara y ihara on discrete subgroups of the two by two projective linear group over p adic fields j math soc japan 19 1966 219 235 many kinds of zeta functions and l functions of a graph or a digraph have been defined and investigated most of the works concerning zeta and l functions of a graph contain the following 1 defining a zeta function 2 defining an l function associated with a regular graph covering 3 providing their determinant expressions and 4 computing the zeta function of a graph covering and obtaining its decomposition formula as a product of l functions as a continuation of those works we introduce a zeta function of a weighted digraph and an l function associated with a weighted digraph bundle a graph bundle is a notion containing a cartesian product of graphs and a regular or irregular graph covering also we provide determinant expressions of the zeta function and the l function moreover we compute the zeta function of a weighted digraph bundle and obtain its decomposition formula as a product of the l functions

independent sets codes and their properties the concept that strict binary relation on free monoids is introduced some characterizations for strict binary relations is given the ordering properties of the set of all strict binary relations as well as some subsets of it are exhibited moreover it is proved that the independent languages of co compatible quasi strict relations are codes

message passing algorithms for counting short cycles in a graph a message passing algorithm for counting short cycles in a graph is presented for bipartite graphs which are of particular interest in coding the algorithm is capable of counting cycles of length g g 2 2g 2 where g is the girth of the graph for a general non bipartite graph cycles of length g g 1 2g 1 can be counted the algorithm is based on performing integer additions and subtractions in the nodes of the graph and passing extrinsic messages to adjacent nodes the complexity of the proposed algorithm grows as o g e 2 where e is the number of edges in the graph for sparse graphs the proposed algorithm significantly outperforms the existing algorithms tailored for counting em short cycles in terms of computational complexity and memory requirements we also discuss a more generic and basic approach of counting short cycles which is based on matrix multiplication and provide a message passing interpretation for such an approach we then demonstrate that an efficient implementation of the matrix multiplication approach has essentially the same complexity as the proposed message passing algorithm

zeta functions of complexes arising from pgl 3 abstract in this paper we obtain a closed form expression of the zeta function z x u of a finite quotient x of the bruhat tits building of pgl 3 over a nonarchimedean local field f by a discrete cocompact torsion free subgroup of pgl 3 analogous to a graph zeta function z x u is a rational function with two different expressions and it satisfies the riemann hypothesis if and only if x is a ramanujan complex

enumeration of graphs with the same ihara zeta function abstract we enumerate all connected graphs with minimal vertex degree 2 on at most 11 vertices and determine their ihara zeta functions we also count the number of such graphs for which there is another graph with the same zeta function we then use these graphs to conjecture properties of the graphs determined by the zeta function in addition we study switching constructions proposed by godsil and mckay to determine whether they preserve the zeta function we show that gm switching is not strong enough to preserve the zeta function but gm switching following the notation of haemers and spence does preserve the zeta function

zeta functions of discrete groups acting on trees abstract this paper generalizes bass work on zeta functions for uniform tree lattices using the theory of von neumann algebras machinery is developed to define the zeta function of a discrete group of automorphisms of a bounded degree tree the main theorems relate the zeta function to determinants of operators defined on edges or vertices of the tree a zeta function associated to a non uniform tree lattice with appropriate hilbert representation is defined zeta functions are defined for infinite graphs with a cocompact or finite covolume group action

the quaternionic second weighted zeta function of a graph and the study determinant abstract we establish a generalization of the second weighted zeta function of a graph to the case of quaternions for an arc weighted graph whose weights are quaternions we define the second weighted zeta function by using the study determinant that is a quaternionic determinant for quaternionic matrices defined by study this definition is regarded as a quaternionic analogue of the determinant expression of hashimoto type for the ihara zeta function of a graph we derive the study determinant expression of bass type and the euler product for the quaternionic second weighted zeta function

the edge l function of a graph abstract we extend watanabe and fukumizu s theorem on the edge zeta function to a regular covering of a graph g next we define an edge l function of a graph g and give a determinant expression of it as a corollary we present a decomposition formula for the edge zeta function of a regular covering of g by a product of edge l functions of g

trace formulae and spectral statistics for discrete laplacians on regular graphs i following the derivation of the trace formulas in the first paper in this series we establish here a connection between the spectral statistics of random regular graphs and the predictions of random matrix theory rmt this follows from the known poisson distribution of cycle counts in regular graphs in the limit that the cycle periods are kept constant and the number of vertices increases indefinitely the result is analogous to the so called diagonal approximation in quantum chaos we also show that by assuming that the spectral correlations are given by rmt to all orders we can compute the leading deviations from the poisson distribution for cycle counts we provide numerical evidence which supports this conjecture

optimization algorithm for reconstructing interface changes of a conductivity inclusion from modal measurements we propose an original optimization approach for reconstructing interface changes of a conductivity inclusion from measurements of eigenvalues and eigenvectors associated with the transmission problem for the laplacian let be a smooth domain and d be an inclusion contained in whose boundary is also assumed to be smooth shape deformation of d causes a perturbation of modal parameters the aim of this work is to show how this information can be used to reconstruct the unknown deformation for doing so we rigorously derive an asymptotic formula for the perturbations in the eigenvalues of the transmission problem for the laplacian that are due to small deformations of the interface of an inclusion based on this formula we design an efficient reconstruction algorithm from modal measurements our algorithm consists on minimizing a functional whose minimizer yields certain geometric properties of the unknown inclusion it naturally follows from a key identity that is in some sense dual to the asymptotic formula numerical experiments showing the viability of our algorithm are presented our results in this paper extend those established in the context of small volume inclusions as well as those for the conductivity interface problem in fact on one hand in a series of recent papers 5 3 1 2 we have derived high order asymptotic expansions of the eigenvalue perturbations due to the presence of small inclusions and used them for locating the inclusions and identifying some of their geometric features on the other hand in 4 we have derived highorder terms in the asymptotic expansions of the boundary perturbations of steady state voltage potentials resulting from small perturbations of the shape of a conductivity inclusion based on these derivations we have designed an effective algorithm to determine some geometric features of the shape perturbation of the inclusion based on boundary measurements

the weighted kirchhoff index of a graph abstract we consider the weighted kirchhoff index of a graph g and present a generalization of somodi s theorem on one of the kirchhoff index of a graph furthermore we give an explicit formula for the weighted kirchhoff index of a regular covering of g in terms of that of g

pseudocodeword performance analysis for ldpc convolutional codes message passing iterative decoders for low density parity check ldpc block codes are known to be subject to decoding failures due to so called pseudocodewords these failures can cause the large signal to noise ratio snr performance of message passing iterative decoding to be worse than that predicted by the maximum likelihood ml decoding union bound in this paper we address the pseudocodeword problem from the convolutional code perspective in particular we compare the performance of ldpc convolutional codes with that of their wrapped quasi cyclic block versions and we show that the minimum pseudoweight of an ldpc convolutional code is at least as large as the minimum pseudoweight of an underlying quasi cyclic code this result which parallels a well known relationship between the minimum hamming weight of convolutional codes and the minimum hamming weight of their quasi cyclic counterparts is due to the fact that every pseudocodeword in the convolutional code induces a pseudocodeword in the block code with pseudoweight no larger than that of the convolutional code s pseudocodeword this difference in the weight spectra leads to improved performance at low to moderate snrs for the convolutional code a conclusion supported by simulation results

reconstructing acoustic obstacles by planar and cylindrical waves in this paper we develop a novel method of reconstructing acoustic obstacles in r2 which follows a similar spirit of the linear sampling method originated by colton and kirsch the reconstruction scheme makes use of the near field measurements encoded into the boundary dirichlet to neumann map or the neumann to dirichlet map both the plane waves and cylindrical waves are shown to meet the reconstruction purpose rigorous mathematical justification of the reconstruction scheme is established the mapping properties of the newly introduced function operators involved in the reconstruction scheme are established these results are of significant mathematical interests for their own sake moreover due to the distinct properties of the function operators the indictor function in the proposed reconstruction scheme exhibits completely different behaviors from those having been established for the indictor function in the original linear sampling method for inverse scattering problems numerical experiments

probing for electrical inclusions with complex spherical waves let a physical body in 2 or 3 be given assume that the electric conductivity distribution inside consists of conductive inclusions in a known smooth background further assume that a subset is available for boundary measurements it is proved using hyperbolic geometry that certain information about the location of the inclusions can be exactly recovered from static electric measurements on more precisely given a ball b with center outside the convex hull of and satisfying b boundary measurements on with explicitly given dirichlet data are enough to determine whether b intersects the inclusion an approximate detection algorithm is introduced based on the theory numerical experiments in dimension two with simulated noisy data suggest that the algorithm finds the inclusion free domain near and is robust against measurement noise 2007 wiley periodicals inc

ruelle zeta functions for finite digraphs abstract three types of expressions for combinatorial zeta functions are considered we investigate conditions for reformulating the exponential expression to the euler product expression and the euler product expression to the determinant expression of hashimoto type the existence of the determinant expression of hashimoto type enables one to unify those zetas within a single framework that is the ruelle zeta functions for quasi finite dynamical systems constructed on finite digraphs which we call the combinatorial zeta functions

the non backtracking spectrum of the universal cover of a graph a non backtracking walk on a graph h is a directed path of directed edges of h such that no edge is the inverse of its preceding edge non backtracking walks of a given length can be counted using the non backtracking adjacency matrix b indexed by h s directed edges and related to ihara s zeta function we show how to determine b s spectrum in the case where h is a tree covering a finite graph we show that when h is not regular this spectrum can have positive measure in the complex plane unlike the regular case we show that outside of b s spectrum the corresponding green function has periodic decay ratios the existence of such a ratio system can be effectively checked and is equivalent to being outside the spectrum we also prove that the spectral radius of the non backtracking walk operator on the tree covering a finite graph is exactly gr where gr is the growth rate of the tree this further motivates the definition of the graph theoretical riemann hypothesis proposed by stark and terras st finally we give experimental evidence that for a fixed finite graph h a ran dom lift of large degree has non backtracking new spectrum near that of h s universal cover this suggests a new generalization of alon s second eigenvalue conjecture

weighted zeta functions of digraphs abstract we define a weighted zeta function of a digraph and a weighted l function of a symmetric digraph and give determinant expressions of them furthermore we give a decomposition formula for the weighted zeta function of a g cyclic cover of a symmetric digraph for any finite group and g a decomposition formula for the weighted zeta function of an oriented line graph l g of a regular covering g of a graph g is given furthermore we define a weighted l function of an oriented line graph l g of g and present a factorization formula for the weighted zeta function of l g by weighted l functions of l g as a corollay we obtain a factorization formula for the multiedge zeta function of g given by stark and terras

codes orderings and partial words codes play an important role in the study of the combinatorics of words in this paper we introduce pcodes that play a role in the study of combinatorics of partial words partial words are strings over a finite alphabet that may contain a number of do not know symbols pcodes are defined in terms of the compatibility relation that considers two strings over the same alphabet that are equal except for a number of insertions and or deletions of symbols we describe various ways of defining and analyzing pcodes in particular many pcodes can be obtained as antichains with respect to certain partial orderings using a technique related to dominoes we show that the pcode property is decidable

distinguishing graphs with zeta functions and generalized spectra abstract conjecturally almost all graphs are determined by their spectra this problem has also been studied for variants such as the spectra of the laplacian and signless laplacian here we consider the problem of determining graphs with ihara and bartholdi zeta functions which are also computable in polynomial time these zeta functions are geometrically motivated but can be viewed as certain generalizations of characteristic polynomials after discussing some graph properties determined by zeta functions we show that large classes of cospectral graphs can be distinguished with zeta functions and enumerate graphs distinguished by zeta functions on 11 vertices this leads us to conjecture that almost all graphs which are not determined by their spectrum are determined by zeta functions along the way we make some observations about the usual types of spectra and disprove a conjecture of setyadi and storm about ihara zeta functions determining degree sequences

backtrackless walks on a graph the aim of this paper is to explore the use of backtrackless walks and prime cycles for characterizing both labeled and unlabeled graphs the reason for using backtrackless walks and prime cycles is that they avoid tottering and can increase the discriminative power of the resulting graph representation however the use of such methods is limited in practice because of their computational cost in this paper we present efficient methods for computing graph kernels which are based on backtrackless walks in a labeled graph and whose worst case running time is the same as that of kernels based on random walks for clustering unlabeled graphs we construct feature vectors using ihara coefficients since these coefficients are related to the frequencies of prime cycles in the graph to efficiently compute the low order coefficients we present an o v 3 algorithm which is better than the o v 6 worst case running time of previously known algorithms in the experimental evaluation we apply the proposed method to clustering both labeled and unlabeled graphs the results show that using backtrackless walks and prime cycles instead of random walks can increase the accuracy of recognition

multilevel linear sampling method for inverse scattering problems a novel multilevel algorithm is presented for implementing the widely used linear sampling method in inverse obstacle scattering problems the new method is shown to possess asymptotically optimal computational complexity for an n times n sampling mesh in mathbb r 2 or an n times n times n sampling mesh in mathbb r 3 the proposed algorithm requires one to solve only mathcal o n n 1 far field equations for a mathbb r n problem n 2 3 and this is in sharp contrast to the original version of the method which needs to solve n n far field equations numerical experiments are presented to illustrate the promising feature of the algorithm in significantly reducing the computational cost of the linear sampling method

zeta functions of line middle total graphs of a graph and their coverings we consider the ihara zeta functions of line graphs middle graphs and total graphs of a regular graph and their regular or irregular covering graphs let l g m g and t g denote the line middle and total graph of g respectively we show that the line middle and total graph of a regular and irregular respectively covering of a graph g is a regular and irregular respectively covering of l g m g and t g respectively for a regular graph g we express the zeta functions of the line middle and total graph of any regular or irregular covering of g in terms of the characteristic polynomial of the covering also the complexities of the line middle and total graph of any regular or irregular covering of g are computed furthermore we discuss the l functions of the line middle and total graph of a regular graph g

zeta functions of several corona type graphs abstract in this paper we present the decomposition formulae for the zeta functions of families of graphs obtained through corona type construction furthermore we construct some families of graphs with the same zeta function by the spectra of these graphs

zeta functions of digraphs abstract we define a zeta function of a digraph and an l function of a symmetric digraph and give determinant expressions of them furthermore we give a decomposition formula for the zeta function of a g cyclic cover of a symmetric digraph for any finite group and g

a generalized bartholdi zeta function for a regular covering of a bipartite graph abstract recently sato 13 introduced a generalized bartholdi zeta function of a bipartite graph and presented three determinant expressions of it at first we present a decomposition formula for the generalized bartholdi zeta function of a regular covering of a bipartite graph g furthermore we introduce a generalized bartholdi l function of g and give three determinant expressions of it as applications we present three types of decomposition formulas for the generalized bartholdi zeta function of a regular covering of g by its generalized bartholdi l functions

local periods and binary partial words an algorithm the study of the combinatorial properties of strings of symbols from a finite alphabet also referred to as words is profoundly connected to numerous fields such as biology computer science mathematics and physics research in combinatorics on words goes back roughly a century there is a renewed interest in combinatorics on words as a result of emerging new application areas such as molecular biology partial words were recently introduced in this context the motivation behind the notion of a partial word is the comparison of genes or proteins alignment of two genes or two proteins can be viewed as a construction of partial words that are said to be compatible while a word can be described by a total function a partial word can be described by a partial function more precisely a partial word of length n over a finite alphabet a is a partial function from 1 n into a elements of 1 n without an image are called holes a word is just a partial word without holes the notion of period of a word is central in combinatorics on words in the case of partial words there arc two notions one is that of period the other is that of local period this paper extends to partial words with one hole the well known result of guibas and odlyzko which states that for every word u there exists a word v of same length as u over the alphabet 0 1 such that the set of all periods of u coincides with the set of all periods of v our result states that for every partial word u with one hole there exists a partial word v of same length as u with at most one hole over the alphabet 0 1 such that the set of all periods of u coincides with the set of all periods of v and the set of all local periods of u coincides with the set of all local periods of v to prove our result we use the technique of halava harju and ilie which they used to characterize constructively the set of periods of a given word as a consequence of our constructive proof we obtain a linear time algorithm which given a partial word with one hole computes a partial word with at most one hole over the alphabet 0 1 with the same length and the same sets of periods and local periods a world wide web server interface at http www uncg edu mat algbin has been established for automated use of the program

ihara zeta functions of digraphs abstract after defining and exploring some of the properties of ihara zeta functions of digraphs we improve upon kotani and sunada s bounds on the poles of ihara zeta functions of undirected graphs by considering digraphs whose adjacency matrices are directed edge matrices

a periodicity result of partial words with one hole abstract the study of the combinatorial properties of strings of symbols from a finite alphabet also referred to as words is profoundly connected to numerous fields such as biology computer science mathematics and physics partial words have long been known they have been around in information and coding theory as well as molecular biology while a word can be described by a total function a partial word can be described by a partial function more precisely a partial word of length n over a finite alphabet a is a partial function from 0 n 1 into a elements of n 1 without an image are called holes a word is just a partial word without holes the notion of period of a word is central in combinatorics on words in the case of partial words there are two notions one is that of period and the other is that of local period there are many fundamental results on periods of words among them is the well known periodicity result of fine and wilf which intuitively determines how far two periodic events have to match in order to guarantee a common period this result was extended to partial words with one hole by berstel and boasson we can define the set per o of all words of maximal length for which fine and wilf s result does not apply similarly we can define the set per 1 of all partial words with one hole of maximal length for which berstel and boasson s result does not apply it is well known that per o contains a unique word up to a renaming that is binary in this note we extend this result to per 1

characteristic polynomials and zeta functions of equitably partitioned graphs abstract let v 1 v r be an equitable partition of the vertex set of a directed graph digraph x it is well known that the characteristic polynomial x x of a quotient graph x divides that of x but the remainder part is not well investigated in this paper we define a deletion graph x over an equitable partition which is a signed directed graph defined for a fixed set of deleting vertices v i v i i 1 r and give a similarity transformation exchanging the adjacency matrix a x which is compatible with the equitable partition for a block triangular matrix whose diagonal blocks are the adjacency matrix of the quotient graph and the deletion graph in fact we show the result for more general matrices including adjacency matrix of graphs and as corollaries we show the followings i a decomposition formula of the reciprocal of the ihara bartholdi zeta function over an equitably partitioned undirected graph into the quotient graph part and the deletion graph part and ii chen and chen s result 4 theorem 3 1 on the ihara bartholdi zeta functions on generalized join graphs and iii teranishi s result 32 theorem 3 3

the enumeration of planar graphs via wick s theorem abstract a seminal technique of theoretical physics called wick s theorem interprets the gaussian matrix integral of the products of the trace of powers of hermitian matrices as the number of labelled maps with a given degree sequence sorted by their euler characteristics this leads to the map enumeration results analogous to those obtained by combinatorial methods in this paper we show that the enumeration of the graphs embeddable on a given 2 dimensional surface a main research topic of contemporary enumerative combinatorics can also be formulated as the gaussian matrix integral of an ice type partition function some of the most puzzling conjectures of discrete mathematics are related to the notion of the cycle double cover we express the number of the graphs with a fixed directed cycle double cover as the gaussian matrix integral of an ihara selberg type function

weighted zeta functions for quotients of regular coverings of graphs abstract let g be a connected graph we reformulate stark and terras galois theory for a quotient h of a regular covering k of a graph g by using voltage assignments as applications we show that the weighted bartholdi l function of h associated to the representation of the covering transformation group of h is equal to that of g associated to its induced representation in the covering transformation group of k furthermore we express the weighted bartholdi zeta function of h as a product of weighted bartholdi l functions of g associated to irreducible representations of the covering transformation group of k we generalize stark and terras galois theory to digraphs and apply to weighted bartholdi l functions of digraphs

graph characteristics from the heat kernel trace graph structures have been proved important in high level vision since they can be used to represent structural and relational arrangements of objects in a scene one of the problems that arises in the analysis of structural abstractions of objects is graph clustering in this paper we explore how permutation invariants computed from the trace of the heat kernel can be used to characterize graphs for the purposes of measuring similarity and clustering the heat kernel is the solution of the heat equation and is a compact representation of the path length distribution on a graph the trace of the heat kernel is given by the sum of the laplacian eigenvalues exponentiated with time we explore three different approaches to characterizing the heat kernel trace as a function of time our first characterization is based on the zeta function which from the mellin transform is the moment generating function of the heat kernel trace our second characterization is unary and is found by computing the derivative of the zeta function at the origin the third characterization is derived from the heat content i e the sum of the elements of the heat kernel we show how the heat content can be expanded as a power series in time and the coefficients of the series can be computed using the laplacian spectrum we explore the use of these characterizations as a means of representing graph structure for the purposes of clustering and compare them with the use of the laplacian spectrum experiments with the synthetic and real world databases reveal that each of the three proposed invariants is effective and outperforms the traditional laplacian spectrum moreover the heat content invariants appear to consistently give the best results in both synthetic sensitivity studies and on real world object recognition problems

zeta equivalent digraphs simultaneous cospectrality abstract we introduce a zeta function of digraphs that determines and is determined by the spectra of all linear combinations of the adjacency matrix its transpose the out degree matrix and the in degree matrix in particular zeta equivalence of graphs encompasses simultaneous cospectrality with respect to the adjacency the laplacian the signless laplacian and the normalized laplacian matrix in addition we express zeta equivalence in terms of markov chains and in terms of invasions where each edge is replaced by a fixed digraph we finish with a method for constructing zeta equivalent digraphs

a trace on fractal graphs and the ihara zeta function starting with ihara s work in 1968 there has been a growing interest in the study of zeta functions of finite graphs by sunada hashimoto bass stark and terras mizuno and sato to name just a few authors then clair and mokhtari sharghi studied zeta functions for infinite graphs acted upon by a discrete group of automorphisms the main formula in all these treatments establishes a connection between the zeta function originally defined as an infinite product and the laplacian of the graph in this article we consider a different class of infinite graphs they are fractal graphs i e they enjoy a self similarity property we de ne a zeta function for these graphs and using the machinery of operator algebras we prove a determinant formula which relates the zeta function with the laplacian of the graph we also prove functional equations and a formula which allows approximation of the zeta function by the zeta functions of finite subgraphs

walk generating functions and spectral measures of infinite graphs let g be a locally finite graph with bounded vertex degrees its adjacency matrix a gives rise to a bounded self adjoint operator on l2 v g let e be the resolution of the identity for a and ev the vector in l2 v g having the v coordinate 1 and all other coordinates 0 the function uv e eu ev is called the spectral measure of g corresponding to vertices u v of g it is related to the generating function wuv z for the number of walks of given length between u and v the spectral measures are used to show the following the pairing theorem characterizing finite bipartite graphs in terms of their spectrum is generalized from finite to infinite graphs the average closed walk generating function can be defined in some cases generalizing the plancherel measure to not necessarily vertex transitive graphs some theorems about the convergence of graph measures are given with applications to the calculation of the expected eigenvalue distribution of a large finite random regular or bipartite semiregular graph and the calculation of spectral measures of several examples

a non commutative formula for the colored jones function the colored jones function of a knot is a sequence of laurent polynomials that encodes the jones polynomial of a knot and its parallels it has been understood in terms of representations of quantum groups and witten gave an intrinsic quantum field theory interpretation of the colored jones function as the expectation value of wilson loops of a 3 dimensional gauge theory the chern simons theory we present the colored jones function as an evaluation of the inverse of a non commutative fermionic partition function this result is in the form familiar in quantum field theory namely the inverse of a generalized determinant our formula also reveals a direct relation between the alexander polynomial and the colored jones function of a knot and immediately implies the extensively studied melvin morton rozansky conjecture first proved by bar natan and the first author about 10 years ago our results complement recent work of huynh and le who also give a non commutative formulae for the colored jones function of a knot starting from a non commutative formula for the r matrix of the quantum group u q mathfrak sl 2 see huynh and le in math gt 0503296

asymptotic imaging of perfectly conducting cracks in this paper we consider cracks with dirichlet boundary conditions we first derive an asymptotic expansion of the boundary perturbations that are due to the presence of a small crack based on this formula we design a noniterative approach for locating a collection of small cracks in order to do so we construct a response matrix from the boundary measurements the location and the length of the crack are estimated respectively from the projection onto the noise space and the first significant singular value of the response matrix indeed the direction of the crack is estimated from the second singular vector we then consider an extended crack with dirichlet boundary conditions we rigorously derive an asymptotic expansion for the boundary perturbations that are due to a shape deformation of the crack to reconstruct an extended crack from many boundary measurements we develop two methods for obtaining a good guess several numerical experiments show how the proposed techniques for imaging small cracks as well as those for obtaining good initial guesses toward reconstructing an extended crack behave

some weighted bartholdi zeta function of a digraph related to the weighted bond scattering matrix of a graph we define some weighted bartholdi zeta function of a digraph d and give a determinant expression for it we present a decomposition formula for the weighted zeta function of a group covering of d furthermore we define an l function of d and give a determinant expression for it as a corollary we express the weighted bartholdi zeta function of a group covering of d by means of its l functions

spectral analysis of the affine graph over the finite ring abstract we apply two methods to the block diagonalization of the adjacency matrix of the cayley graph defined on the affine group the affine group will be defined over the finite ring z p n z the method of irreducible representations will allow us to find nontrivial eigenvalue bounds for two different graphs one of these bounds will result in a family of ramanujan graphs the method of covering graphs will be used to block diagonalize the affine graphs using a galois group of graph automorphisms in addition we will demonstrate the method of covering graphs on a generalized version of the graphs of lubotzky et al a lubotzky r phillips p sarnak ramanujan graphs combinatorica 8 1988 261 277

the coefficients of the immanantal polynomial abstract an expression of the coefficient of immanantal polynomial of an n n matrix is present moreover we give expressions of the coefficient of immanantal polynomials of combinatorial matrices adjacency matrix laplacian matrix signless laplacian matrix as applications we show that the immanantal polynomials for laplacian matrix and signless laplacian matrix of bipartite graphs are the same this is a generalization of the characteristic polynomial for laplacian matrix and signless laplacian matrix of bipartite graphs furthermore we consider the relations between the characteristic polynomial and the immanantal polynomial for trees

strengthened linear sampling method with a reference ball by adding a reference ball as an extra artificial obstacle component to the underlying scattering system we propose a simple but robust and effective technique to choose the crucial cut off value required in the linear sampling method lsm for inverse scattering problems the reference ball technique causes few extra computational costs to the lsm but brings in a practically important byproduct i e it eliminates the interior eigenvalue problem automatically a well known barrier when applying the lsm some mathematical justifications of the new technique are provided and numerical experiments are also presented to demonstrate its feasibility and effectiveness

zeta functions of finite graphs and coverings iii abstract a graph theoretical analog of brauer siegel theory for zeta functions of number fields is developed using the theory of artin l functions for galois coverings of graphs from parts i and ii in the process we discuss possible versions of the riemann hypothesis for the ihara zeta function of an irregular graph

laplacian immanantal polynomials and the gts poset on trees abstract let t be a tree on n vertices with laplacian l t and let gts n be the generalised tree shift poset on the set of unlabelled trees on n vertices inequalities are known for coefficients of the characteristic polynomial of l t as we go up the poset gts n in this work we generalise these inequalities to the q laplacian l t q of t and to the coefficients of all immanantal polynomials

partial words and a theorem of fine and wilf revisited a word of length n over a finite alphabet a is a map from 0 n 1 into a a partial word of length n over a is a partial map from 0 n 1 into a in the latter case elements of 0 n1 without image are called holes a word is just a partial word without holes in this paper we extend a fundamental periodicity result on words due to fine and wilf to partial words with two or three holes this study was initiated by berstel and boasson for partial words with one hole partial words are motivated by molecular biolog

eigenvalue monotonicity of q laplacians of trees along a poset abstract let t be a tree on n vertices with q laplacian matrix l t q let gts n be the generalized tree shift poset on the set of unlabelled trees with n vertices we prove that for all q r going up on gts n has the following effect the spectral radius of l t q increases while the smallest eigenvalue of l t q decreases we also prove that for all q r with q 1 going up on gts n increases the second smallest eigenvalue of l t q these generalize known results for eigenvalues of the laplacian matrix of trees as a corollary we obtain consequences about the eigenvalues of q t laplacians and exponential distance matrices of trees

the critical ising model via kac ward matrices the kac ward formula allows to compute the ising partition function on any finite graph g from the determinant of 22g matrices where g is the genus of a surface in which g embeds we show that in the case of isoradially embedded graphs with critical weights these determinants have quite remarkable properties first of all they satisfy some generalized kramers wannier duality there is an explicit equality relating the determinants associated to a graph and to its dual graph also they are proportional to the determinants of the discrete critical laplacians on the graph g exactly when the genus g is zero or one finally they share several formal properties with the ray singer overline partial torsions of the riemann surface in which g embeds

dependences related to strict binary relations in this paper we study dependences on a free monoid x and their relations to strict binary relations on x as a consequence we characterize strict binary relations by the notion of dependence moreover some results concerning the relationship between dependences and codes are also obtained a visualization of this work is provided by the theory of graph

kac ward formula and its extension to order disorder correlators through a graph zeta function a streamlined derivation of the kac ward formula for the planar ising model s partition function is presented and applied in relating the kernel of the kac ward matrices inverse with the correlation functions of the ising model s order disorder correlation functions a shortcut for both is facilitated by the bowen lanford graph zeta function relation the kac ward relation is also extended here to produce a family of non planar interactions on mathbb z 2 for which the partition function and the order disorder correlators are solvable at special values of the coupling parameters temperature

on the exponential generating function for non backtracking walks we derive an explicit formula for the exponential generating function associated with non backtracking walks around a graph we study both undirected and directed graphs our results allow us to derive computable expressions for non backtracking versions of network centrality measures based on the matrix exponential we find that eliminating backtracking walks in this context does not significantly increase the computational expense we show how the new measures may be interpreted in terms of standard exponential centrality computation on a certain multilayer network insights from this block matrix interpretation also allow us to characterize centrality measures arising from general matrix functions rigorous analysis on the star graph illustrates the effect of non backtracking and shows that unwanted localization effects can be eliminated when we restrict to non backtracking walks we also investigate the localization issue on synthetic networks

an ihara formula for partially directed graphs abstract ihara s formula expresses the ihara zeta function of a finite undirected graph as a rational function with a particularly nice form in 2001 mizuno and sato showed that the ihara zeta function of a fully directed graph has a similar expression and in 2005 sato generalized ihara s formula to connected simple partially directed graphs sato proved his formula for the more general two variable bartholdi zeta function this paper provides a new proof of ihara s formula for the ihara zeta function of any finite graph not necessarily connected or simple no matter whether it is undirected fully directed or partially directed

quasi cyclic ldpc codes influence of proto and tanner graph structure on minimum hamming distance upper bounds quasi cyclic qc low density parity check ldpc codes are an important instance of proto graph based ldpc codes in this paper we present upper bounds on the minimum hamming distance of qc ldpc codes and study how these upper bounds depend on graph structure parameters like variable degrees check node degrees girth of the tanner graph and of the underlying proto graph moreover for several classes of proto graphs we present explicit qc ldpc code constructions that achieve or come close to the respective minimum hamming distance upper bounds because of the tight algebraic connection between qc codes and convolutional codes we can state similar results for the free hamming distance of convolutional codes in fact some qc code statements are established by first proving the corresponding convolutional code statements and then using a result by tanner that says that the minimum hamming distance of a qc code is upper bounded by the free hamming distance of the convolutional code that is obtained by unwrapping the qc code

periodicity on partial words a partial word of length n over a finite alphabet a is a partial map from 0 n 1 into a elements of 0 n 1 without image are called holes a word is just a partial word without holes a fundamental periodicity result on words due to fine and wilf 1 intuitively determines how far two periodic events have to match in order to guarantee a common period this result was extended to partial words with one hole by berstel and boasson 2 and to partial words with two or three holes by blanchet sadri and hegstrom 3 in this paper we give an extension to partial words with an arbitrary number of holes

hook immanantal and hadamard inequalities for q laplacians of trees abstract let t be a tree on n vertices with laplacian matrix l and q laplacian l q let k be the character of the irreducible representation of s n indexed by the hook partition k 1 n k and let d k l be the normalized hook immanant of l corresponding to the character k inequalities for d k l as k increases are known by assigning a statistic to vertex orientations on trees we generalize these inequalities to the q analogue l q of l for all q r and to the bivariate q t laplacian l q t for some values q t our statistic based approach also generalizes several other inequalities including the changing index k l of the hadamard inequality for l to the matrix l q and l q t thus we extend several results about l to l q which includes the case when l q is not positive semidefinite

identification of small inhomogeneities asymptotic factorization we consider the boundary value problem of calculating the electrostatic potential for a homogeneous conductor containing finitely many small insulating inclusions we give a new proof of the asymptotic expansion of the electrostatic potential in terms of the background potential the location of the inhomogeneities and their geometry as the size of the inhomogeneities tends to zero such asymptotic expansions have already been used to design direct i e noniterative reconstruction algorithms for the determination of the location of the small inclusions from electrostatic measurements on the boundary e g music type methods our derivation of the asymptotic formulas is based on integral equation methods it demonstrates the strong relation between factorization methods and music type methods for the solution of this inverse problem

conjugacy on partial words the study of the combinatorial properties of strings of symbols from a finite alphabet also referred to as words is profoundly connected to numerous fields such as biology computer science mathematics and physics in this paper we examine to which extent some fundamental combinatorial properties of words such as conjugacy remain true for partial words the motivation behind the notion of a partial word is the comparison of two genes alignment of two such strings can be viewed as a construction of two partial words that are said to be compatible this study on partial words was initiated by berstel and boasson