the minimum rank of symmetric matrices described by a graph a survey abstract the minimum rank of a simple graph g is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry for i j is nonzero whenever i j is an edge in g and is zero otherwise this paper surveys the current state of knowledge on the problem of determining the minimum rank of a graph and related issues

techniques for determining the minimum rank of a small graph abstract the minimum rank of a simple graph g is defined to be the smallest possible rank over all symmetric real matrices whose ij th entry for i j is nonzero whenever i j is an edge in g and is zero otherwise minimum rank is a difficult parameter to compute however there are now a number of known reduction techniques and bounds that can be programmed on a computer we have developed a program using the open source mathematics software sage to implement several techniques we have also established several additional strategies for computation of minimum rank these techniques have been used to determine the minimum ranks of all graphs of order 7

multiplicity results for double phase problems in rn we consider a double phase problem in rn with a q 1 superlinear subcritical caratheodory reaction term which does not satisfy the ambrosetti rabinowitz condition using variational methods based on the topological degree and critical point theory together with the nehari manifold method and deformation lemma we find three nontrivial solutions among them the first one is the positive ground state solution the second one is the negative ground state solution and the third one is the least energy sign changing solution which changes sign only once we also discuss the existence of infinitely many solutions for even energy functional and that of radial solutions

on the possible multiplicities of the eigenvalues of a hermitian matrix whose graph is a tree we consider the general problem of determining which lists of multiplicities for the eigenvalues occur among hermitian matrices the graph of whose off diagonal entries is a given tree several restrictions are cited and a construction strategy is given together these are sufficient to characterize all lists for each tree in two infinite classes the double paths and generalized stars and to tabulate all lists for trees on fewer than nine vertices such tables should be useful for formulating and dispelling general conjectures

a note on universally optimal matrices and field independence of the minimum rank of a graph abstract for a simple graph g on n vertices the minimum rank of g over a field f written as mr f g is defined to be the smallest possible rank among all n n symmetric matrices over f whose i j th entry for i j is nonzero whenever i j is an edge in g and is zero otherwise a symmetric integer matrix a such that every off diagonal entry is 0 1 or 1 is called a universally optimal matrix if for all fields f the rank of a over f is the minimum rank of the graph of a over f recently dealba et al l m dealba j grout l hogben r mikkelson k rasmussen universally optimal matrices and field independence of the minimum rank of a graph electron j linear algebra 18 2009 403 419 initiated the study of universally optimal matrices and established field independence or dependence of minimum rank for some families of graphs in the present paper more results on universally optimal matrices and field independence or dependence of the minimum rank of a graph are presented and some results of dealba et al 5 are improved

the maximum corank of graphs with a 2 separation abstract for a graph g v e with vertex set v 1 2 n which is allowed to have parallel edges and for a field f let s g f be the set of all f valued symmetric n n matrices a which represent g the maximum corank of a graph g is the maximum possible corank over all a s g f if g 1 g 2 is a 2 separation we give a formula which relates the maximum corank of g to the maximum corank of some small variations of g 1 and g 2

computation of minimal rank and path cover number for certain graphs for a given undirected graph g the minimum rank of g is defined to be the smallest possible rank over all real symmetric matrices a whose i j th entry is non zero whenever i j and i j is an edge in g the path cover number of g is the minimum number of vertexdisjoint paths occurring as induced subgraphs of g that cover all the vertices of g for trees the relationship between minimum rank and path cover number is completely understood however for non trees only sporadic results are known we derive formulae for the minimum rank and path cover number for graphs obtained from edge sums and formulae for minimum rank of vertex sums of graphs in addition we examine previously identified special types of vertices and attempt to unify the theory behind them

minimum rank and path cover number for generalized and double generalized cycle star graphs abstract for a given connected undirected graph g v e with v 1 n the minimum rank of g is defined to be the smallest possible rank over all symmetric matrices a a i j such that for i j a i j 0 if and only if i j e the path cover number of g is the minimum number of vertex disjoint paths occurring as induced subgraphs of g that cover all the vertices of g when g is a tree the values of the minimum rank and of the path cover number are known as well the relationship between them we study these values and their relationship for all graphs that have at most two vertices of degree greater than two generalized cycle stars and double generalized cycle stars

decompositions of minimum rank matrices let f be a field let g be an undirected graph on n vertices and let sf g be the class of all f valued symmetric n n matrices whose nonzero off diagonal entries occur in exactly the positions corresponding to the edges of g for each graph g there is an associated minimum rank class mrf g consisting of all matrices a sf g with ranka mrf g for most graphs g with connectivity 1 or 2 we give explicit decompositions of matrices in mrf g into sums of minimum rank matrices of simpler graphs usually proper subgraphs related to g our results can be thought of as generalizations of well known formulae for the minimum rank of a graph with a cut vertex and of a graph with a 2 separation we conclude by also showing that for these graphs matrices in mrf g can be constructed from matrices of simpler graphs moreover we give analogues for positive semidefinite matrices

minimum rank matrices with prescribed graph abstract we study properties of real symmetric matrices with prescribed graph and lowest possible rank we concentrate on the case where the graph is a tree and give an algorithm for computing the minimum possible rank in this case a more general problem is also considered

on the difference between the maximum multiplicity and path cover number for tree like graphs abstract for a given undirected graph g the maximum multiplicity of g is defined to be the largest multiplicity of an eigenvalue over all real symmetric matrices a whose i j th entry is non zero whenever i j and i j is an edge in g the path cover number of g is the minimum number of vertex disjoint paths occurring as induced subgraphs of g that cover all the vertices of g we derive a formula for the path cover number of a vertex sum of graphs and use it to prove that the vertex sum of so called non deficient graphs is non deficient for unicyclic graphs we provide a complete description of the path cover number and the maximum multiplicity and hence the minimum rank and we investigate the difference between path cover number and maximum multiplicity for a class of graphs referred to as block cycle graphs

vertex and edge spread of zero forcing number maximum nullity and minimum rank of a graph abstract the minimum rank of a simple graph g is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry for i j is nonzero whenever i j is an edge in g and is zero otherwise maximum nullity is taken over the same set of matrices the zero forcing number is the minimum size of a zero forcing set of vertices and bounds the maximum nullity from above the spread of a graph parameter at a vertex v or edge e of g is the difference between the value of the parameter on g and on g v or g e rank spread at a vertex was introduced in 4 this paper introduces vertex spread of the zero forcing number and edge spreads for minimum rank maximum nullity and zero forcing number properties of the spreads are established and used to determine values of the minimum rank maximum nullity and zero forcing number for various types of grids with a vertex or edge deleted

zero forcing sets and the minimum rank of graphs the minimum rank of a simple graph g is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry for i 6 j is nonzero whenever i j is an edge in g and is zero otherwise this paper introduces a new graph parameter z g that is the minimum size of a zero forcing set of vertices and uses it to bound the minimum rank for numerous families of graphs often enabling computation of the minimum rank

on minimum rank and zero forcing sets of a graph abstract for a graph g on n vertices and a field f the minimum rank of g over f written as mr f g is the smallest possible rank over all n n symmetric matrices over f whose i j th entry for i j is nonzero whenever ij is an edge in g and is zero otherwise the maximum nullity of g over f is m f g n mr f g the minimum rank problem of a graph g is to determine mr f g or equivalently m f g this problem has received considerable attention over the years in f barioli w barrett s butler s m cioab d cvetkovic s m fallat c godsil w haemers l hogben r mikkelson s narayan o pryporova i sciriha w so d stevanovic h van der holst k v meulen a w wehe aim minimum rank special graphs work group zero forcing sets and the minimum rank of graphs linear algebra appl 428 2008 1628 1648 a new graph parameter z g the zero forcing number was introduced to bound m f g from above the authors posted an attractive question what is the class of graphs g for which z g m f g for some field f this paper focuses on exploring the above question

on the multiplicity of the adjacency eigenvalues of graphs abstract let g be a simple graph with the adjacency matrix a g a well known result of cvetkovic and gutman states that removing a pendant vertex and its neighbour does not change the nullity of the graph we generalize this theorem and some other theorems similar to it for an arbitrary eigenvalue of a graph also for an arbitrary eigenvalue we use a corresponding star set to delete some subgraphs and to determine the multiplicity of we use star sets to find some parter wiener vertices in trees we state our results for real symmetric matrices and so for weighted graphs and as a corollary we use them for simple graphs furthermore we use these methods to obtain some results for 0 1

the minimum rank of matrices and the equivalence class graph abstract for a given connected undirected graph g the minimum rank of g v g e g is defined to be the smallest possible rank over all hermitian matrices a whose i j th entry is non zero if and only if i j and i j is an edge in g i j e g for each vertex x in g x v g n x is the set of all neighbors of x let r be the equivalence relation on v g such that x y v g xry n x n y our aim is find classes of connected graphs g v g e g such that the minimum rank of g is equal to the number of equivalence classes for the relation r on v g

the inverse inertia problem for graphs cut vertices trees and a counterexample abstract let g be an undirected graph on n vertices and let s g be the set of all real symmetric n n matrices whose nonzero off diagonal entries occur in exactly the positions corresponding to the edges of g the inverse inertia problem for g asks which inertias can be attained by a matrix in s g we give a complete answer to this question for trees in terms of a new family of graph parameters the maximal disconnection numbers of a graph we also give a formula for the inertia set of a graph with a cut vertex in terms of inertia sets of proper subgraphs finally we give an example of a graph that is not inertia balanced which settles an open problem from the october 2006 aim workshop on spectra of families of matrices described by graphs digraphs and sign patterns we also determine some restrictions on the inertia set of any graph

on the minimum rank of the join of graphs and decomposable graphs abstract for a given undirected graph g the minimum rank of g is defined to be the smallest possible rank over all real symmetric matrices a whose i j th entry is nonzero whenever i j and i j is an edge in g in this work we consider joins and unions of graphs and characterize the minimum rank of such graphs in the case of balanced inertia several consequences are provided for decomposable graphs also known as cographs

on the relative position of multiple eigenvalues in the spectrum of an hermitian matrix with a given graph for hermitian matrices whose graph is a given tree the relationships among vertex degrees multiple eigenvalues and the relative position of the underlying eigenvalue in the ordered spectrum are discussed in detail in the process certain aspects of special vertices whose removal results in an increase in multiplicity are investigated

intensive structure of heisenburg supergenerator algebras over hv su2 dual carrier subspaces in liouville nmr formalisms superbosons as unit tensor superoperators abstract the structure of supergenerator algebra over simply reducible carrier space h v associated with a lie algebra over the domain of tensorial bases of nmr density operator formalisms for spin coherence are set out in terms of s 2 i s i s j s 2 j or s 2 i superbosonic forms corresponding to six unit tensors or fundamental wigner superoperators the kernel defining the characteristic null space of these unit tensors is also given some extensions of these concepts to the realm of many body nmr spin cluster problems and to rota like invariants over dual spin space are summarise the s 2 i set of component s bosons are shown to generate a group algebra and to display the requisite ladder aspects of the i set of superoperators in this respect the underlying premise yields an improved insight into the sets of su2 and su2 x s n tensors associated with nmr formalisms for spin dynamics or for model realizations of coherence transfer or relaxation by providing the intensive substructure of liouville carrier space the mapping aspects within which it has been shown that each h v is a simply reducible carrier subspace is seen as an essential preliminary aspect of su2 duality over liouville space in understanding spin cluster systems in nmr the value of p kq s 2 i monomial formalisms in appreciating the structure of liouville space and its associated transformational properties is stressed

on the minimum rank of the third power of a starlike tree abstract for a simple graph g of order n let a be a real n n symmetric matrix whose i j th entry for i j is nonzero whenever i j is an edge in g and is zero otherwise the minimum rank of g is the smallest possible rank over all such symmetric matrices the jth power of a graph g is the graph g j v f where u v f if and only if there is a walk of length j from u to v in 2007 brualdi hogben and shader reported a conjecture that if t is not the star k 1 n 1 then mr t 3 mr t 2 1 in this paper we construct a class of starlike trees such that mr t 3 mr t 2 which give a negative answer to this conjecture

contrasting routes to the kostka coefficients of λ boxvr n ln permutational module expansions for 6 n 8 applications to nmr spin clusters under su m n ln spin symmetries abstract aspects of the higher n l n permutational modules associated with young subgroups of various highly branched high n fold algebras which are pertinent to identical spin nmr clusters are presented for boxvr n or boxvr n arota p tuple or number partition the method of optimal choice for deriving the kostka coefficients found in sets derived from permutational module expansions rests on the ordering of the shape to the self associated diagram s in the dominance hierarchy hence physical insight into these cage cluster nmr systems is developed both from these properties and from the inter related induced symmetries of gl n and l n groups from these associated combinatorial mapping or scalar invariant aspects of su m n l n symmetry one may define the a n l n systems of ax n nmr problems in a general semi topological limit this corresponds to a high n l n limit in which the individual spin cluster exhibits a lack of any intracluster magnetic equivalence properties

an upper bound for the minimum rank of a graph abstract for a graph g of order n the minimum rank of g is defined to be the smallest possible rank over all real symmetric n n matrices a whose i j th entry for i j is nonzero whenever i j is an edge in g and is zero otherwise we prove an upper bound for minimum rank in terms of minimum degree of a vertex is valid for many graphs including all bipartite graphs and conjecture this bound is true over for all graphs and prove a related bound for all zero nonzero patterns of not necessarily symmetric matrices most of the results are valid for matrices over any infinite field but need not be true for matrices over finite fields

the minimum rank of a sign pattern matrix with a 1 separation abstract a sign pattern matrix is a matrix whose entries are from the set 0 if a is an m n sign pattern matrix the qualitative class of a denoted q a is the set of all real m n matrices b b i j with b i j positive respectively negative zero if a i j is respectively 0 the minimum rank of a sign pattern matrix a denoted mr a is the minimum of the ranks of the real matrices in q a determination of the minimum rank of a sign pattern matrix is a longstanding open problem for the case that the sign pattern matrix has a 1 separation we present a formula to compute the minimum rank of a sign pattern matrix using the minimum ranks of certain generalized sign pattern matrices associated with the 1 separation

minimum rank and maximum eigenvalue multiplicity of symmetric tree sign patterns abstract the set of real matrices described by a sign pattern a matrix whose entries are elements of 0 has been studied extensively but only loose bounds were available for the minimum rank of a tree sign pattern a simple graph has been associated with the set of symmetric matrices having a zero nonzero pattern of off diagonal entries described by the graph and the minimum rank maximum eigenvalue multiplicity among matrices in this set is readily computable for a tree in this paper we extend techniques for trees to tree sign patterns and trees allowing loops with the presence or absence of loops describing the zero nonzero pattern of the diagonal allowing precise computation of the minimum rank of a tree sign pattern and a tree allowing loops for a symmetric tree sign pattern or a tree that allows loops we provide an algorithm that allows exact computation of maximum multiplicity and minimum rank and can be used to obtain a symmetric integer matrix realizing minimum rank

spectral multiplicity and splitting results for a class of qualitative matrices abstract this paper presents some results concerning the location and multiplicity of eigenvalues of sign symmetric matrices whose associated graphs are trees in particular it extends previous spectral multiplicity and splitting results proved by others

the minimum rank problem over the finite field of order 2 minimum rank 3 our main result is a sharp bound for the number of vertices in a minimal forbidden subgraph for the graphs having minimum rank at most 3 over the finite field of order 2 we also list all 62 such minimal forbidden subgraphs we conclude by exploring how some of these results over the finite field of order 2 extend to arbitrary fields and demonstrate that at least one third of the 62 are minimal forbidden subgraphs over an arbitrary field for the class of graphs having minimum rank at most 3 in that field

computing inertia sets using atoms we consider the problem of computing inertia sets for graphs by using tools for combining the inertia sets of smaller graphs we can reduce this problem to understanding the inertia sets for three connected graphs that are not joins we term such graphs atoms and give the inertia sets for all atoms on at most seven vertices this can be used to compute the inertia sets for all graphs on at most seven vertices

orthogonal representations minimum rank and graph complements orthogonal representations are used to show that complements of certain sparse graphs have positive semidefinite minimum rank at most 4 this bound applies to the complement of a 2 tree and to the complement of a unicyclic graph hence for such graphs the sum of the minimum rank of the graph and the minimum rank of its complement is at most two more than the order of the graph the minimum rank of the complement of a 2 tree is determined exactly

on the maximum multiplicity of an eigenvalue in a matrix whose graph contains exactly one cycle we consider the general problem of determining the maximum possible multiplicity of an eigenvalue in a hermitian matrix whose graph contains exactly one cycle for some cases we express that maximum multiplicity in terms of certain parameters associated with the graph

on acyclic and unicyclic graphs whose minimum rank equals the diameter the minimum rank of a graph g is defined as the smallest possible rank over all symmetric matrices governed by g it is well known that the minimum rank of a connected graph is at least the diameter of that graph in this paper we investigate the graphs for which equality holds between minimum rank and diameter and completely describe the acyclic and unicyclic graphs for which this equality holds

minimum rank problems abstract a graph describes the zero nonzero pattern of a family of matrices with the type of graph undirected or directed simple or allowing loops determining what type of matrices symmetric or not necessarily symmetric diagonal entries free or constrained are described by the graph the minimum rank problem of the graph is to determine the minimum among the ranks of the matrices in this family the determination of maximum nullity is equivalent this problem has been solved for simple trees p m nylen minimum rank matrices with prescribed graph linear algebra appl 248 1996 303 316 c r johnson a leal duarte the maximum multiplicity of an eigenvalue in a matrix whose graph is a tree linear and multilinear algebra 46 1999 139 144 trees allowing loops l m dealba t l hardy i r hentzel l hogben a wangsness minimum rank and maximum eigenvalue multiplicity of symmetric tree sign patterns linear algebra appl 418 2006 389 415 and directed trees allowing loops f barioli s fallat d hershkowitz h t hall l hogben h van der holst b shader on the minimum rank of not necessarily symmetric matrices a preliminary study electron j linear algebra 18 2000 126 145 we survey these results from a unified perspective and solve the minimum rank problem for simple directed trees

minimum rank of outerplanar graphs abstract the problem of finding the minimum rank over all symmetric matrices corresponding to a given graph has grown in interest recently it is well known that the minimum rank of any graph is bounded above by the clique cover number the minimum number of cliques needed to cover all edges of the graph we generalize the idea of the clique cover number by defining the rank sum of a cover to be the sum of the minimum ranks of the graphs in the cover using this idea we obtain a combinatorial solution to the minimum rank problem for an outerplanar graph as a consequence the minimum rank of an outerplanar graph is field independent and all outerplanar graphs have a universally optimal matrix we also consider implications of the main result to the inverse inertia problem

maximum nullity of outerplanar graphs and the path cover number abstract let g v e be a graph with v 1 2 n define s g as the set of all n n real valued symmetric matrices a a ij with a ij 0 i j if and only if ij e by m g we denote the largest possible nullity of any matrix a s g the path cover number of a graph g denoted p g is the minimum number of vertex disjoint paths occurring as induced subgraphs of g which cover all the vertices of g there has been some success with relating the path cover number of a graph to its maximum nullity johnson and duarte 5 have shown that for a tree t m t p t barioli et al 2 show that for a unicyclic graph g m g p g or m g p g 1 notice that both families of graphs are outerplanar we show that for any outerplanar graph g m g p g further we show that for any partial 2 path g m g p g

forbidden minors for the class of graphs g with ξ g 2 abstract for a given simple graph g s g is defined to be the set of real symmetric matrices a whose i j th entry is nonzero whenever i j and ij is an edge in g in f barioli s fallat l hogben a variant on the graph parameters of colin de verdiere implications to the minimum rank of graphs electron j linear algebra 13 2005 387 404 g is defined to be the maximum corank i e nullity among a s g having the strong arnold property is used to study the minimum rank maximum eigenvalue multiplicity problem for g since is minor monotone the graphs g such that g k can be described by a finite set of forbidden minors we determine the forbidden minors for g 2 and present an application of this characterization to computation of minimum rank among matrices in s g

zero forcing parameters and minimum rank problems the zero forcing number z g which is the minimum number of vertices in a zero forcing set of a graph g is used to study the maximum nullity minimum rank of the family of symmetric matrices described by g it is shown that for a connected graph of order at least two no vertex is in every zero forcing set the positive semidefinite zero forcing number z g is introduced and shown to be equal to g os g where os g is the recently defined ordered set number that is a lower bound for minimum positive semidefinite rank the positive semidefinite zero forcing number is applied to the computation of positive semidefinite minimum rank of certain graphs an example of a graph for which the real positive symmetric semidefinite minimum rank is greater than the complex hermitian positive semidefinite minimum rank is presented