new bounds for the extreme values of a finite sample of real numbers we derive new bounds for the smallest valuexminand the largest valuexmaxof a finite samplex1 xnof real numbers our bounds are obtained by solving two optimization problems one of them being convex and the other nonconvex we show that the pair xmin xmax lies in a region bounded by an ellipse and an hyperbola the corresponding cartesian equations are given in terms of the average and the standard deviation of the sample

extremal hexagonal chains with respect to the coefficients sum of the permanental polynomial a hexagonal system is a graphical representation of a benzenoid hydrocarbon in theoretical chemistry a hexagonal chain is a cata condensed hexagonal system with no branchings in this paper we consider extremal hexagonal chains with respect to the coefficients sum of the permanental polynomial we prove that the linear chain attains the minimum value of this sum and the zigzag chain attains the maximum value of this sum

new limits to classical and quantum instance compression given an instance of a hard decision problem a limited goal is to compress that instance into a smaller equivalent instance of a second problem as one example consider the problem where given

isomorphisms of adjoint chevalley groups over integral domains it is shown that every automorphism of an adjoint chevalley group over an integral domain containing the rational number field is uniquely expressible as the product of a ring automorphism a graph automorphism and an inner automorphism while every isomorphism between simple adjoint chevalley groups can be expressed uniquely as the product of a ring isomorphism a graph isomorphism and an inner automorphism the isomorphisms between the elementary subgroups are also found having analogous expressions

on the greatest distance between two permanental roots of a matrix abstract the permanental spread of a complex square matrix a is defined to be the greatest distance between two roots of the equation per zi a 0 a preliminary study of this number as well as of two related quantities is given in particular we derive upper and lower bounds and deal with comparisons of different bounds finally two inequalities involving the permanental spread are treated

on the accuracy of the gerschgorin circle theorem for bounding the spread ofa symmetric matrix abstract the spread of a matrix with real eigenvalues is the difference between its largest and smallest eigenvalues the gerschgorin circle theorem can be used to bound the extreme eigenvalues of the matrix and hence its spread for nonsymmetric matrices the gerschgorin bound on the spread may be larger by an arbitrary factor than the actual spread even if the matrix is symmetrizable this is not true for real symmetric matrices it is shown that for real symmetric matrices or complex hermitian matrices the ratio between the bound and the spread is bounded by p 1 where p is the maximum number of off diagonal nonzeros in any row of the matrix for full matrices this is just n this bound is not quite sharp for n greater than 2 but examples with ratios of n 1 for all n are given for banded matrices with m nonzero bands the maximum ratio is bounded by m independent of the size of n this bound is sharp provided only that n is at least 2m for sparse matrices p may be quite small and the gerschgorin bound may be surprisingly accurate

on the characterizing properties of the permanental polynomials of graphs abstract let g be a graph and a g the adjacency matrix of g the permanental polynomial of g is defined as g x per x i a g if two graphs g and h have the same permanental polynomial then g is called a per cospectral mate of h a graph g is said to be characterized by its permanental polynomial if all the per cospectral mates of g are isomorphic to g it is shown that complete graphs stars regular complete bipartite graphs and odd cycles are characterized by their permanental polynomials we prove that in general the permanental polynomial cannot characterize the paths and even cycles in particular for each l 1 and m 2 we can find non isomorphic per cospectral mates of p 4 l 3 and c 4 m respectively when we restrict our consideration to connected graphs both the paths and even cycles c 4 l 2 are characterized by their permanental polynomials

some inequalities for central moments of matrices abstract in this paper we shall study non commutative central moment inequalities with a focus on whether the commutative bounds are tight in the non commutative case as well we prove that the answer is affirmative for the fourth central moment and several particular results are given in the general case as an application we shall present some lower estimates of the spread of hermitian and normal matrices as well

automorphisms and isomorphisms of chevalley groups and algebras abstract an adjoint chevalley group of rank at least 2 over a rational algebra or a similar ring its elementary subgroup and the corresponding lie ring have the same automorphism group these automorphisms are explicitly described

resolvent expansions of matrices and applications abstract various explicit expansions of the resolvent of a square complex matrix in a neighborhood of the origin including the well known laurent expansion are obtained simple proofs using algebraic arguments rather than the theory of complex functions are given cesaro average expansions of the resolvent on the spectral circle are also developed several applications of these expansions are given in particular these include some new inverse nonnegativity properties of the drazin inverses of m matrices

bounds for the spectrum of normal matrices abstract let a be a normal matrix with eigenvalues 1 2 n and let denote the smallest disc containing these eigenvalues we give some inequalities relating the center and radius of to the entries in a when applied to hermitian matrices our results give lower bounds on the spread maxij i j of a when applied to positive definite hermitian matrices they give lower bounds on the kantorovich ratio maxij i j i j

designing fpt algorithms for cut problems using randomized contractions we introduce a new technique for designing fixed parameter algorithms for cut problems called randomized contractions we apply our framework to obtain the first fixed parameter algorithms fpt algorithms with exponential speed up for the steiner cut and node multiway cut uncut problems we prove that the parameterized version of the unique label cover problem which is the base of the unique games conjecture can be solved in 2 o k 2 log sigma n 4 log n deterministic time even in the stronger vertex deletion variant where k is the number of unsatisfied edges and sigma is the size of the alphabet as a consequence we show that one can in polynomial time solve instances of unique games where the number of edges allowed not to be satisfied is upper bounded by o sqrt log n to optimality which improves over the trivial o 1 upper bound we prove that the steiner cut problem can be solved in 2 o k 2 log k n 4 log n deterministic time and tilde o 2 o k 2 log k n 2 random

expansions of sums of matrix powers it is shown that the n fold partial sums of powers of a square complex matrix having spectral radius one or less can be approximated by matrix polynomials where the approximation is taken in the sense of cesaro averages the degree of the approximating polynomials and the order of cesaro averaging is characterized when the matrix is nonnegative the results are strengthened and it is shown that cesaro limits can be replaced by regular limits of periodic subsequences the periods of these expansions are characterized we obtain the explicit computable form of all approximating polynomials generalizing known results for stochastic and irreducible nonnegative matrices several applications of the results are discussed

upper bounds on the maximum modulus of subdominant eigenvalues of nonnegative matrices abstract upper bounds for the maximum modulus of the subdominant roots of square nonnegative matrices are obtained we provide a unified approach that yields or improves upon most of the bounds that have been obtained so far

satisfiability allows no nontrivial sparsification unless the polynomial time hierarchy collapses consider the following two player communication process to decide a language l the first player holds the entire input x but is polynomially bounded the second player is computationally unbounded but does not know any part of x their goal is to decide cooperatively whether x belongs to l at small cost where the cost measure is the number of bits of communication from the first player to the second player for any integer d 3 and positive real e we show that if satisfiability for n variable d cnf formulas has a protocol of cost o nd e then conp is in np poly which implies that the polynomial time hierarchy collapses to its third level the result even holds when the first player is conondeterministic and is tight as there exists a trivial protocol for e 0 under the hypothesis that conp is not in np poly our result implies tight lower bounds for parameters of interest in several areas namely sparsification kernelization in parameterized complexity lossy compression and probabilistically checkable proofs by reduction similar results hold for other np complete problems for the vertex cover problem on n vertex d uniform hypergraphs this statement holds for any integer d 2 the case d 2 implies that no np hard vertex deletion problem based on a graph property that is inherited by subgraphs can have kernels consisting of o k2 e edges unless conp is in np poly where k denotes the size of the deletion set kernels consisting of o k2 edges are known for several problems in the class including vertex cover feedback vertex set and bounded degree deletion

permanental polynomials of graphs abstract the first section surveys recent results on the permanental polynomial of a square matrix a i e per xi a the second section concerns the permanental polynomial of the adjacency matrix of a graph the final section is an introduction to the permanental polynomial of the laplacian matrix of a graph an appendix lists some of these latter polynomials

polyhedral object recognition by indexing in computer vision the indexing problem is the problem of recognizing a few objects in a large database of objects while avoiding the help of the classical image feature to object feature matching paradigm in this paper we address the problem of recognizing three dimensional 3 d polyhedral objects from 2 d images by indexing both the objects to be recognized and the images are represented by weighted graphs the indexing problem is therefore the problem of determining whether a graph extracted from the image is present or absent in a database of model graphs we introduce a novel method for performing this graph indexing process which is based both on polynomial characterization of binary and weighted graphs and on hashing we describe in detail this polynomial characterization and then we show how it can be used in the context of polyhedral object recognition next we describe a practical recognition by indexing system that includes the organization of the database the representation of polyhedral objects in terms of 2 d characteristic views the representation of this views in terms of weighted graphs and the associated image processing finally some experimental results allow the evaluation of the system performance

kernel bounds for disjoint cycles and disjoint paths in this paper we show that the problems disjoint cycles and disjoint paths do not have polynomial kernels unless np conp poly thus these problems do not allow polynomial time preprocessing that results in instances whose size is bounded by a polynomial in the parameter at hand we build upon recent results by bodlaender et al 6 and fortnow and santhanam 20 that show that np complete problems that are or compositional do not have polynomial kernels unless np conp poly to this machinery we add a notion of transformation and obtain that disjoint cycles and disjoint paths do not have polynomial kernels unless np conp poly for the proof we introduce a problem on strings called disjoint factors and first show that this problem has no polynomial kernel unless np conp poly we also show that the related disjoint cycles packing problem has a kernel of size o klogk

enumeration of permanental sums of lattice graphs abstract the permanental sum ps g is the coefficients sum of the permanental polynomial of a graph g in this work we exhibit the explicit expressions of the permanental sums of quadrilateral linear chains on the plane and quadrilateral linear rings on the cylinder by establishing orientation graphs respectively meanwhile we show that the permanental sum of the hexagonal linear chain with n hexagons is n 1 2 which determines the value of permanental sum of the extremal hexagonal chain in 9

laplacian spread of graphs lower bounds and relations with invariant parameters abstract the spread of an n n complex matrix b with eigenvalues 1 2 n is defined by s b max i j i j where the maximum is taken over all pairs of eigenvalues of b let g be a graph on n vertices the concept of laplacian spread of g is defined by the difference between the largest and the second smallest laplacian eigenvalue of g in this work by combining old techniques of interlacing eigenvalues and rank 1 perturbation matrices new lower bounds on the laplacian spread of graphs are deduced some of them involving invariant parameters of graphs as it is the case of the bandwidth independence number and vertex connectivity

on the geometry and laplacian of a graph abstract let g be a simple graph on n vertices and let l be the laplacian matrix of g we point out some connections between the geometric properties of g and the spectrum of l the multiplicities and eigenspaces as well as the eigenvalues of l are of geometric interest some historical information and relations of l to other matrices associated with g are also described

divide and conquer approximation algorithms via spreading metrics we present a novel divide and conquer paradigm for approximating np hard graph optimization problems the paradigm models graph optimization problems that satisfy two properties first a divide and conquer approach is applicable second a fractional spreading metric is computable in polynomial time the spreading metric assigns lengths to either edges or vertices of the input graph such that all subgraphs for which the optimization problem is nontrivial have large diameters in addition the spreading metric provides a lower bound t on the cost of solving the optimization problem we present a polynomial time approximation algorithm for problems modeled by our paradigm whose approximation factor is o min log t log log t log k log log k where k denotes the number of interesting vertices in the problem instance and is at most the number of vertices we present seven problems that can be formulated to fit the paradigm for all these problems our algorithm improves previous results the problems are 1 linear arrangement 2 embedding a graph in a d dimensional mesh 3 interval graph completion 4 minimizing storage time product 5 subset feedback sets in directed graphs and multicuts in circular networks 6 symmetric multicuts in directed networks 7 balanced partitions and p separators for small values of p in directed graphs

parameterized graph separation problems we consider parameterized problems where some separation property has to be achieved by deleting as few vertices as possible the following five problems are studied delete k vertices such that a each of the given l terminals is separated from the others b each of the given l pairs of terminals is separated c exactly l vertices are cut away from the graph d exactly l connected vertices are cut away from the graph e the graph is separated into at least l components we show that if both k and l are parameters then a b and d are fixed parameter tractable while c and e are w 1 hard

some inequalities for positive linear maps it has long been known that an analogue of jensen s inequality holds for positive unital linear maps on matrix algebras provided that instead of ordinary convex functions one restricts to matrix convex functions we show that this restriction is not necessary in the case of 2 2 matrices a noncommutative analogue of the variance is studied and a basic inequality with several applications is established

upper bounds on the laplacian spread of graphs abstract the laplacian spread of a graph g is defined as the difference between the largest and the second smallest eigenvalue of the laplacian matrix of g in this work an upper bound for this graph invariant that depends on first zagreb index is given moreover another upper bound is obtained and expressed as a function of the nonzero coefficients of the laplacian characteristic polynomial of a graph

fixed parameter tractability of multicut parameterized by the size of the cutset given an undirected graph g a collection s 1 t 1 dots s k t k of pairs of vertices and an integer p the edge multicut problem asks if there is a set s of at most p

an algorithm for the second immanant let x be an irreducible character of the symmetric group s for a a j an n by n matrix define the immanant of a corresponding to x by d a e x gf ha lt ugsn t 1 the article contains an algorithm for computing d a when x corresponds to the partition 2 n 2 introduction denote by xk the irreducible characteristic zero character of the symmetric group sn corresponding to the partition k in k for k 1 2 n if a ai1 is an n by n matrix define n dk a xk uf h at t a s t l then for example dl a det a and dn a per a the permanent of a in general dk is known as an immanant or a generalized matrix function an immanant is a generalized matrix function based on sn suppose g is a simple graph on n vertices denote by l g the laplacian matrix corresponding to some labeling of the vertices of g i e l g is an n by n matrix the i j entry of which is the degree of vertex i when i j 1 if i 7 j but vertex i is adjacent to vertex j and zero otherwise it is shown in 5 that the number of hamiltonian circuits in g is given by the formula i n k 1 h g 1 dk l g 2nk 2 while there is an immense literature on generalized matrix functions eq 1 is already sufficient motivation to seek fast algorithms for their actual computation the main result of this note is an algorithm for computing d2 see the next section it seems that d2 may be especially appropriate for the study of laplacian matrices for the following reason if g is a graph on n vertices then l g is positive semidefinite symmetric and singular moreover g is connected if and only if rank l g n 1 for arbitrary positive semidefinite symmetric matrices without a zero row it was established in 3 corollaries 5 and 6 that d2 a 0 with equality if and only if rank a n 1 received august 22 1983 1980 mathematics subject classification primary 65f30 15a15 05c50 work of the second author was supported by the national science foundation under grant no mcs 8300097 01984 american mathematical society 0025 5718 84 1 00 25 per page

using gauss jordan elimination to compute the index generalized nullspaces and drazin inverse abstract we develop and analyze a new algorithm that computes bases for the null spaces of all powers of a given matrix as well as its index the algorithm uses row operations and shuffling steps in which rows of pairs of matrices are interchanged in particular the new algorithm may be viewed as an extension of the classic gauss jordan elimination method for inverting a nonsingular matrix it is also shown that the drazin inverse has a simple representation in terms of the output of the algorithm and the original matrix

immanantal invariants of graphs abstract something between an expository note and an extended research problem this article is an invitation to expand the existing literature on a family of graph invariants rooted in linear and multilinear algebra there are a variety of ways to assign a real n n matrix k g to each n vertex graph g so that g and h are isomorphic if and only if k g and k h are permutation similar it follows that g and h are isomorphic only if k g and k h are similar i e that similarity invariants of k g are graph theoretic invariants of g an observation that helps to explain the enormous literature on spectral graph theory the focus of this article is the permutation part i e on matrix functions that are preserved under permutation similarity if not under all similarity

meta kernelization in a parameterized problem every instance i comes with a positive integer k the problem is said to admit a polynomial kernel if in polynomial time one can reduce the size of the instance i to a polynomial in k while preserving the answer in this work we give two meta theorems on kernelization the first theorem says that all problems expressible in counting monadic second order logic and satisfying a coverability property admit a polynomial kernel on graphs of bounded genus our second result is that all problems that have finite integer index and satisfy a weaker coverability property admit a linear kernel on graphs of bounded genus these theorems unify and extend all previously known kernelization results for planar graph problems

the diameter and width of numerical ranges abstract we characterize the diameter and width of the numerical range present an algorithm for computing the diameter and width of the numerical range and formulate the diameters of the numerical ranges of unitary bordering matrices for lower dimensions we also determine the boundaries of the numerical ranges of certain nilpotent toeplitz matrices to be curves of constant width

minimizing the least eigenvalues of unicyclic graphs with application to spectral spread the spread of a graph is defined to be the difference between the largest eigenvalue and the least eigenvalue of the adjacency matrix of the graph let u k denote the set of connected unicyclic graphs of order n and girth k and let un denote the set of connected unicyclic graphs of order n in this paper we determine the unique graph with minimum least eigenvalue respectively the unique graph with maximum spread among all graphs in u k we finally characterize the unique graph with minimum least eigenvalue respectively the unique graph with maximum spread among all graphs in un

bounds on distances between eigenvalues explicit computable lower and upper bounds on the distances between a given real eigenvalue of a real square matrix and the remaining not necessarily real eigenvalues of the matrix are developed

on the permanental sum of graphs abstract let g be a graph and a g the adjacency matrix of g the polynomial g x per x i a g is called the permanental polynomial of g and the permanental sum of g is the summation of the absolute values of the coefficients of g x in this paper we investigate properties of permanental sum of a graph prove recursive formulas to compute the permanental sum of a graph and show that the ordering of graphs with respect to permanental sum furthermore we determine the upper and lower bounds of permanental sum of unicyclic graphs and the corresponding extremal unicyclic graphs are also determined

on the bivariate permanent polynomials of graphs abstract in 1980 balasubramanian and parthasarathy introduced the bivariate permanent polynomials of graphs and conjectured that this polynomial is a graph characterizing polynomial that is any two graphs with the same bivariate permanent polynomial are isomorphic in this paper we give counterexamples to this conjecture by a computer search furthermore we show that several well known families of graphs are determined by the bivariate permanent polynomial complete graphs complete bipartite graphs regular complete multipartite graphs cycles and their complements

automorphisms of chevalley groups of different types over commutative rings abstract in this paper we prove that every automorphism of elementary adjoint chevalley group with root system of rank 1 over a commutative ring with 1 2 for the systems a 2 f 4 b l c l with 1 2 and 1 3 for the system g 2 is standard i e it is a composition of ring inner central and graph automorphisms

automorphisms of chevalley groups of type f4 over local rings with 1 2 abstract in the given paper we prove that every automorphism of a chevalley group of type f 4 over a commutative local ring with 1 2 is standard i e it is a composition of ring and inner automorphisms

on the structure of stochastic matrices with a subdominant eigenvalue near 1 abstract an n n irreducible stochastic matrix p can possess a subdominant eigenvalue say 2 p near 1 in this article we clarify the relationship between the nearness of these eigenvalues and the near uncoupling some authors say nearly completely decomposable of p we prove that for fixed n if 2 p is sufficiently close to 1 then p is nearly uncoupled we then provide examples which show that 2 p must in general be remarkably close to 1 before such uncoupling occurs

a bound for the permanent of the laplacian matrix abstract it is shown that if g is a simple connected graph on n vertices then per l g 2 n 1 g where l g is the laplacian matrix of g and g is the complexity of g

per spectral characterizations of graphs with extremal per nullity abstract a graph g is said to be determined by its permanental spectrum if any graph having the same permanental spectrum as g is isomorphic to g in this paper we introduce the permanental nullity of a graph the multiplicity of the number zero in the permanental spectrum of a graph to study graphs determined by their permanental spectra first we determine all graphs of order n whose permanental nullities are n 2 n 3 n 4 and n 5 respectively then we show that all graphs with the permanental nullity n 2 n 3 or n 5 and all non bipartite graphs with the permanental nullity n 4 are determined by their permanental spectra in particular we prove that the complete bipartite graphs are determined by their permanental spectra

representative sets and irrelevant vertices new tools for kernelization we continue the development of matroid based techniques for kernelization initiated by the present authors 47 we significantly extend the usefulness of matroid theory in kernelization by showing applications of a result on representative sets due to lovasz 51 and marx 53 as a first result we show how representative sets can be used to derive a polynomial kernel for the elusive almost 2 sat problem where the task is to remove at most k clauses to make a 2 cnf formula satisfiable solving a major open problem in kernelization this result also yields a new o log opt approximation for the problem improving on the o log n approximation of agarwal et al 3 and an implicit o log opt approximation due to even et al 24 we further apply the representative sets tool to the problem of finding irrelevant vertices in graph cut problems that is vertices that can be made undeletable without affecting the answer to the problem this gives the first significant progress towards a polynomial kernel for the multiway cut problem in particular we get a kernel of o ks 1 vertices for multiway cut instances with at most s terminals both these kernelization results have significant spin off effects producing the first polynomial kernels for a range of related problems more generally the irrelevant vertex results have implications for covering min cuts in graphs for a directed graph g v e and sets s t v let r be the size of a minimum s t vertex cut which may intersect s and t we can find a set z v of size o s t r that contains a minimum a b vertex cut for every a s b t similarly for an undirected graph g v e a set of terminals x v and a constant s we can find a set z v of size o x s 1 that contains a minimum multiway cut for every partition of x into at most s pairwise disjoint subsets both results are polynomial time we expect this to have further applications in particular we get direct reduction rule based kernelizations for all problems above in contrast to the indirect compression based kernel previously given for odd cycle transversal 47 all our results are randomized with failure probabilities that can be made exponentially small in n due to needing a representation of a matroid to apply the representative sets tool

the aα spread of a graph abstract let a g and d g be the adjacency matrix and the degree diagonal matrix of a graph g respectively for any real number 0 1 nikiforov defined the a matrix of a graph g as a g d g 1 a g the a spread of a graph is the difference between the largest eigenvalue and the smallest eigenvalue of the a matrix of the graph in this paper we obtain some new lower bounds on a spread and characterize the extremal graphs for certain cases which extend the results of a spread and q spread moreover some graph operations on a spread are presented

lower bounds for the spread of a matrix a characterization of the spread of a normal matrix is used to derive several simple lower bounds for the spread comparisons are then made with several known bounds

eigenvalues of a h generalized join graph operation constrained by vertex subsets a generalized h join operation of a family of graphs g1 gp where h has order p constrained by a family of vertex subsets si v gi i 1 p is introduced when each vertex subset si is ki i regular it is deduced that all non main adjacency eigenvalues of gi different from ki i remain as eigenvalues of the graph g obtained by this operation if each gi is ki regular and all the vertex subsets are such that si v gi the h generalized join constrained by these vertex subsets coincides with the h join operation furthermore some applications on the spread of graphs are presented

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

lower bounds for the spread of a hermitian matrix abstract some simple lower bounds for the spread of a hermitian matrix are derived these bounds are explicit functions of the entries of the matrix one of which is sharper than a recent result due to e r barnes and a j hoffman comparisons are made with several known results

positive linear maps and perturbation bounds of matrices abstract we show how positive unital linear maps can be used to obtain lower bounds for the maximum distance between the eigenvalues of two normal matrices some related bounds for the spread and condition number of hermitian matrices are also discussed here

characterizations and lower bounds for the spread of a normal matrix abstract the spread of an n n matrix a with eigenvalues 1 n is defined by spr a max j k j k we prove that if a is normal then spr a max x ax y ay x y c n x y 1 max x ay y ax x y c n x y 1 re x y 0 max x ay y ax x y c n x y 1 x y 0 max spr z a z a 2 z c z 1 2 max x a 2 x x ax 2 x a ax x ax 2 1 2 x c n x 1 we also present several lower bounds for spr a given by these characterizations

the laplacian spread of quasi tree graphs abstract the laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the second smallest eigenvalue of the laplacian matrix of the graph bao tan and fan y h bao y y tan y z fan the laplacian spread of unicyclic graphs appl math lett 22 2009 1011 1015 characterize the unique unicyclic graph with maximum laplacian spread among all connected unicyclic graphs of fixed order in this paper we characterize the unique quasi tree graph with maximum laplacian spread among all quasi tree graphs in the set q n d with 1 d n 4 2

computing the permanental polynomials of graphs let m be an nn matrix with entries mij i j 1 2 n the permanent of m is defined to be per m i 1nmi i where the sum is taken over all permutations of 1 2 n the permanental polynomial of m is defined by per xinm where in is the identity matrix of size n in this paper we give recursive formulas for computing permanental polynomials of the laplacian matrix and the signless laplacian matrix of a graph respectively

unicyclic graphs with second largest and second smallest permanental sums abstract let a g be an adjacency matrix of a graph g then the polynomial g x per x i a g is called the permanental polynomial of g and the permanental sum of g is the sum of the absolute values of the coefficients of g x in this paper the second largest and second smallest permanental sums among connected unicyclic graphs and the corresponding extremal graphs are determined in addition we show that the computation of permanental sum is p complete

new bounds for the signless laplacian spread abstract let g be an undirected simple graph the signless laplacian spread of g is defined as the maximum distance of pairs of its signless laplacian eigenvalues this paper establishes some new bounds both lower and upper for the signless laplacian spread several of these bounds depend on invariant parameters of the graph we also use a minmax principle to find several lower bounds for this spectral invariant

permanental roots and the star degree of a graph abstract for undirected graphs without loops or multiple edges we define the star degree of a graph and prove that it is equal to the multiplicity of the root 1 of per xi b where b d a considering bipartite graphs we prove that per xi b per xi l where l d a and consequently that the star degree of a bipartite graph can also be characterized by the multiplicity of the root 1 of per xi l

multiplicity of integer roots of polynomials of graphs abstract let g be a graph with minimum vertex degree p 1 let b d a where d is the diagonal matrix of vertex degrees and a is the adjacency matrix of g the multiplicity of the integer root p of per xi b is characterized for bipartite graphs this characterization extends to per xi l where l d a is the laplacian matrix of g for graphs with unrestricted vertex degrees bounds are obtained on the multiplicities of integer roots of the permanental and characteristic polynomials of both l and b

antidiagonals of matrices in a unitary similarity orbit abstract let mn be the algebra of n n complex matrices for a aij mn the kth antidiagonal of a is the vector ak 1 ak 1 2 a1 k t and u a u au u is unitary is the unitary similarity orbit of a we study the sets 1 d k a the collection of all kth antidiagonals of matrices in u a 2 i k a the collection of z k i 1 v i c such that v v 1 v k t d k a 3 p k a the collection of z k i 1 v i c such that v v 1 v k t d k a particular attention is paid to the relation between the geometrical properties of the sets and the algebraic properties of the matrix a when k 1 all three sets reduce to w a the numerical range of a which has been studied extensively we also consider the following related concepts 1 d k a max x x d k a where x x x 1 2 2 s k a max z z i k a 3 p k a max z z p k a

half integrality lp branching and fpt algorithms a recent trend in parameterized algorithms is the application of polytope tools to fixed parameter tractable fpt algorithms e g cygan et al focs 2011 52nd annual symposium on foundations of computer science ieee 2011 pp 150 159 narayanaswamy et al stacs 2012 symposium on theoretical aspects of computer science 2012 pp 338 349 although this approach has yielded significant speedups for a range of important problems it requires the underlying polytope to have very restrictive properties including half integrality and nemhauser trotter style persistence properties to date these properties are essentially known to hold only for two classes of polytopes covering the cases of vertex cover nemhauser and trotter math program 8 1975 pp 232 248 and node multiway cut garg et al j alg 50 2004 pp 49 61 taking a slightly different approach we view half integrality as a discrete relaxation of a problem e g a relaxation of the search space from 0 1 v to 0

highly unique network descriptors based on the roots of the permanental polynomial in this paper we examine the zeros of permanental polynomials as highly unique network descriptors we employ exhaustively generated networks and demonstrate that our defined graph measures based on the moduli of the zeros of permanental polynomials are quite efficient when distinguishing graphs structurally in this work we continue with a line of research that relates to the search of almost complete graph invariants these highly unique network measures may serve as a powerful tool for tackling graph isomorphism

almost all trees are co immanantal abstract if t is a tree denote by a t its adjacency matrix and by l t its laplacian matrix write d2 for the second immanant matrix function the main result is this let tn be the number of trees on n vertices let sn be the number of such trees t for which there is a nonisomorphic tree t such that d2 xi a t d2 xi a t and d2 xi l t d2 xi l t then sn tn 1 as n