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

diameters and eigenvalues we derive a new upper bound for the diameter of a k regular graph g as a function of the eigenvalues of the adjacency matrix namely suppose the adjacency matrix of g has eigenvalues ai a2 a with ia i i vi21 ia i where ai k a ia21 then the diameter d g must satisfy d g we will consider families of graphs whose eigenvalues can be explicitly de termined these graphs are determined by sums or differences of vertex labels namely the pair i j being an edge depends only on the value i j or i j for directed graphs we will show that these graphs are expander graphs with small diameters by using an inequality on character sums which was recently proved by n m katz bell communications research 445 south street morristown new jersey 07960 this content downloaded from 207 46 13 111 on sat 21 may 2016 07 00 04 utc all use subject to http about jstor org terms

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

finite analogues of euclidean space graphs are attached to fqn where fq is the field with q elements q odd using an analogue of the euclidean distance the graphs are shown to be asymptotically ramanujan for large q better than ramanujan in half the cases comparisons are made with finite upper half planes constructed in a similar way using an analogue of poincare s non euclidean distance the eigenvalues of the adjacency operators of the finite euclidean graphs are shown to be kloosterman sums

sturm liouville eigenproblems with an interior pole the eigenvalues and eigenfunctions of self adjoint sturm liouville problems with a simple pole on the interior of the interval a b are investigated three general theorems are proved and it is shown that as n the eigenfunctions more and more closely resemble those of an ordinary sturm liouville problem and n m2 2 b a 2 just as if there were no singularity the low order modes however differ drastically from those of a nonsingular eigenproblem in that i both eigenvalues and eigenfunctions are complex despite the fact the problem is self adjoint ii the real and imaginary parts of the nth eigenfunction may both have ever increasing numbers of interior zeros as b instead of just n 1 zeros and iii as b the eigenvalues for all small n may cluster about a common value in contrast to the widely separated eigenvalues of the corresponding nonsingular problem these results are general but in order to present quantitative solutions for the low order modes too special attention is gi

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

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

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

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

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

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

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

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

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

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

relative positions of limit cycles in a kolmogorov type system to estimate the relative position of limit cycles for a system is always useful in the qualitative analysis of the system if the system has only one limit cycle determining the position of the limit cycle is more important the author constructs an annular region containing all the limit cycles for a kolmogorov type system since the system contains the lotka volterra gausse kuang and freedman and huang models as special cases the results obtained are also valid for those models furthermore the approach to the location of limit cycles is very applicable because the region is easy to compute

dissipative eigenvalue problems for a sturm liouville operator with a singular potential in this paper we consider the sturm liouville operator d 2 dx 2 1 x on the interval a b a 0 f x f x gammaf 0 with gamma is an element of c boolean or r or by the dirichlet condition f 0 f 0 0 we also show that the corresponding operators can be obtained by norm resolvent approximation from operators where the potential 1 x is replaced by a continuous function and that their eigen and associated functions can be chosen to form a bari basis in l 2 a b

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

graph spectra for finite upper half planes over rings abstract graphs attached for finite analogues of the poincare upper half plane over rings z p r z are introduced for p an odd prime the spectra of these graphs for r 2 are shown to be related to those for r 1 which were studied earlier in contrast to the case r 1 we find that the graphs are not ramanujan graphs when r 2 and p 5 histograms of the eigenvalues for r 1 and 2 are also compared the graphs over rings are of interest for connections with p adic upper half planes as well as fundamental domains for congruence subgroups of the modular group sl 2 z

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 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

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

on the coulomb potential in one dimension a mathematically rigorous definition of the one dimensional schrodinger operator d 2 dx 2 gamma x is given it is proven that the domain of the operator is defined by the boundary conditions connecting the values of the function on the left and right half axes the investigated operator is compared with the schrodinger operator containing the coulomb potential gamma x

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

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

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

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

the linearization of boundary eigenvalue problems and reproducing kernel hilbert spaces the boundary eigenvalue problems for the adjoint of a symmetric relation s in a hilbert space with finite not necessarily equal defect numbers which are related to the selfadjoint hilbert space extensions of s are characterized in terms of boundary coefficients and the reproducing kernel hilbert spaces they induce 2001 elsevier science inc all rights reserved

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

response to comment on penetrability of a one dimensional coulomb potential by roger g newton for original paper see ibid vol 26 p 2445 1993 by expressing in matrix form the one dimensional hamiltonians where the potentials are proportional to mod x mod 1 and x 1 i show that we are dealing with two very different problems thus in my viewpoint the criticism of r g newton of the paper mentioned in the title which is based on results relating to mod x mod 1 does not apply to a potential of the form x 1

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 note on uniqueness of limit cycles in a lienard type system abstract 5 proposed a theorem to guarantee the uniqueness of limit cycles for the generalized lienard system dx dt h y f x dy dt g x we will give a counterexample to their theorem it will be shown that their theorem is valid only if f x is monotone on certain intervals for this case we give a shorter proof and we also show that the limit cycle is hyperbolic if the condition on the monotonicity of f x is violated then we need an additional condition to guarantee the uniqueness of the limit cycle examples are provided to illustrate our results

uniqueness of limit cycles in a lienard type system abstract in this paper conditions that guarantee the uniqueness of limit cycles in the lienard type system dx dt h y f x dy dt g x have been discussed the conditions obtained here are different from those of zhang and others several examples such as q x 0 are given to illustrate that our results may easily be employed in practical applications

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 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

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

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

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

comment on penetrability of a one dimensional coulomb potential by m moshinsky see ibid vol 26 p 2445 1993 the analysis of the one dimensional schrodinger equation with the potential lambda x given is found to be incorrect contrary to the author s claim the origin is indeed impenetrable

limit cycles in a general two stroke oscillation abstract conditions of the existence of limit cycles in a general two stroke oscillation system are obtained the general oscillation is a special kind of lienard type system with isoclines approaching both positive infinites or negative infinites as t the conditions for the uniqueness of the two stroke oscillation are also proved some examples are used to illustrate our results

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

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

perturbation theory for a sturm liouville problem with an interior singularity the perturbation theory of operators and forms is used to construct sturm liouville differential operators for potentials with 1 x pf 1 x and for epsilon rightarrow 0 1 x i epsilon interior singularities the norm resolvent convergence of approximating sequences of operators with smooth potentials is established and various qualitative and quantitative properties of their spectra are obtained for an infinite interval a comparison is made with the one dimensional coulomb problem possessing a potential 1 x

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