on the relation between the positive inertia index and negative inertia index of weighted graphs abstract let g w be an edge weighted graph and let a g w be its adjacency matrix the positive inertia index respectively the negative inertia index of g w denoted by p g w resp p g w is defined to be the number of positive eigenvalues resp negative eigenvalues of a g w clearly p g w p g w is the rank of g w whereas p g w p g w is defined as the signature of g w in this paper some upper and lower bounds on p g w p g w are determined for the weighted chordal graphs resp bipartite graphs complete graphs cactus graphs join graphs bisplit graphs power graphs

an upper bound of the nullity of a graph in terms of order and maximum degree abstract let g be a finite undirected graph without loops and multiple edges by and n we respectively denote the nullity the maximum vertex degree and the order of g in 10 it was proved that n 2 n 1 when g is a tree this result was generalized to a bipartite graph by 25 for a reduced bipartite graph g the above inequality was improved to n 2 2 ln 2 by 27 however the problem of bounding the nullity of an arbitrary graph g in terms of n and is left open for more than ten years in this article we aim to solve such a left problem we prove that 1 n for an arbitrary graph g with order n and maximum degree and the equality holds if and only if g is the disjoint union of some copies of k

on the nullity of the line graph of unicyclic graph with depth one abstract a connected graph with a unique cycle is called a unicyclic graph a unicyclic graph with depth one may be thought of as being obtained from a cycle by appending n i pendent edges on each vertex v i in the cycle c t for some integer t 3 denoted by c n 1 n 2 n t in this paper we give a complete characterization on the nullity of the line graph g l c n 1 n 2 n t as follows let m i n i 0 then i g 2 if and only if m 0 and t 0 mod 4 ii g 1 if and only if m 1 and either 1 n i 0 1 for i 1 t the length of any zero chain of n 1 n 2 n t is even and t m 0 mod 4 or 2 t 0 mod 4 and one of n 1 n 3 n t 1 0 and n 2 n 4 n t 0 must hold iii g 0 otherwise

the maximum fuzzy weighted matching models and hybrid genetic algorithm a matching of a graph is an independent subset of the edges set and a maximum matching is a matching with as many edges in it as possible the maximum weighted matching problem is to find a maximum matching in a given graph such that the sum of the weights of the edges in it is maximum in this paper the concepts of expected maximum fuzzy weighted matching the maximum fuzzy weighted matching and the most maximum fuzzy weighted matching are initialized according to various decision criteria the maximum fuzzy weighted matching problem is formulated as expected value model chance constrained programming and dependent chance programming by using the credibility theory and the crisp equivalents are also given furthermore a hybrid genetic algorithm is designed for solving the proposed fuzzy programming models finally a numerical example is given

a characterization of graphs g with nullity v g 2m g 2c g abstract by v g e g g and m g we denote respectively the order the number of edges the nullity and the matching number of a simple graph g recently wang and wong have proved that for every graph g g v g 2 m g 2 c g where c g e g v g g g being the number of connected components of g in this paper graphs g that satisfy the equality g v g 2 m g 2 c g are characterized

on the nullity of a graph with cut points let g be a simple graph of order n and a g be its adjacency matrix the nullity of a graph g denoted by g is the multiplicity of the eigenvalue zero in the spectrum of a g denote by ck and lk the set of all connected graphs with k induced cycles and the set of line graphs of all graphs in ck respectively in 1998 sciriha i sciriha on singular line graphs of trees congr numer 135 1998 73 91 show that the order of every tree whose line graph is singular is even then gutman and sciriha i gutman i sciriha on the nullity of line graphs of trees discrete math 232 2001 35 45 show that the nullity set of l0 is 0 1 in this paper we investigate the nullity of graphs with cut points and deduce some concise formulas then we generalize scirihas result showing that the order of every graph g is even if such a graph g satisfies that g ck and l g k 1 and the nullity set of lk is 0 1 k k 1 for any given k where l g denotes the line graph of the graph g

proof of a conjecture on the nullity of a connected graph in terms of order and maximum degree abstract let g be a connected graph with n vertices maximum vertex degree 2 and nullity in q zhou et al 2018 22 it was conjectured that 2 n 2 1 and the upper bound is attained if and only if g is a cycle c n with n divisible by 4 or a complete bipartite graph k n 2 n 2 in the present paper this conjecture is proved

upper bound of skew energy of an oriented graph in terms of its skew rank abstract let g be an oriented graph with skew adjacency matrix s g the skew energy e s g of g is the sum of the norms of all eigenvalues of s g and the skew rank r s g of g is the rank of s g recently tian and wong gave a lower bound of e s g in terms of its skew rank they proved that r s g e s g and they characterized the oriented graphs which satisfy the equality in this paper we aim to establish an upper bound for skew energy of an oriented graph in terms of its skew rank and maximum vertex degree it is proved that e s g r s g for an arbitrary oriented graph g with maximum degree and the upper bound is attained if and only if g is the disjoint union of r s g 2 copies of k together with some isolated vertices where the orientation is switching equivalent to the elementary orientation of k which assigns all edges the same direction from a chromatic set to another one

a characterization of graphs with rank 5 abstract the rank of a graph g is defined to be the rank of its adjacency matrix in this paper we consider the following problem what is the structure of a connected graph g with rank 5 or equivalently what is the structure of a connected n vertex graph g whose adjacency matrix has nullity n 5 in this paper we completely characterize connected graphs g whose adjacency matrix has rank 5

the multiplicity of an arbitrary eigenvalue of a graph in terms of cyclomatic number and number of pendant vertices abstract let g be a graph with adjacency matrix a g the nullity g of g is the multiplicity of zero as an eigenvalue of a g which has received a lot of attention because of its chemical importance the multiplicity of an arbitrary eigenvalue of a g is denoted as m g in 25 the authors proved that g 2 g p g for a connected graph g with equality if and only if g is a cycle of order divisible by 4 where g e g v g 1 is the cyclomatic number of g and p g is the number of pendant vertices of g in the present paper we intend to extend this result from the nullity of g to the multiplicity of an arbitrary eigenvalue of g differing from the method in 25 by mainly applying algebraic method we prove the following result for a connected graph g m g 2 g p g for an arbitrary eigenvalue of g the equality holds if and only if g is a cycle c n and 2 cos 2 k n with k 1 2 n 2 1

on the nullity of bipartite graphs abstract the nullity of a graph is defined to be the multiplicity of the eigenvalue zero in the spectrum of the adjacency matrix of the graph in this paper we obtain the nullity set of bipartite graphs of order n and characterize the bipartite graphs with nullity n 4 and the regular bipartite graphs with nullity n 6

trees with maximum nullity abstract the nullity of a graph is the multiplicity of the eigenvalue zero in its spectrum among n vertex trees the star has greatest nullity equal to n 2 we generalize this by showing that among n vertex trees whose vertex degrees do not exceed a certain value d the greatest nullity is n 2 n 1 d methods for constructing such maximum nullity trees are described

a sharp upper bound of the nullity of a connected graph in terms of order and maximum degree abstract the nullity g of g is the multiplicity of 0 as an eigenvalue of a g in this paper we completely solve the following conjecture proposed by zhou wong and sun in linear algebra and its applications 555 2018 314 320 let g be a connected graph of order n with nullity g and the maximum degree 2 then g 2 n 2 1 the equality holds if and only if g c n n 0 m o d 4 or g k

the evans krylov theorem for nonlocal parabolic fully nonlinear equations abstract in this paper we prove the evans krylov theorem for nonlocal parabolic fully nonlinear equations

matching and symmetry of graphs abstract matching is a mathematical concept that deals with the way of spanning a given graph network with a set of pairs of adjacent points it is pointed out that in many different areas of science and culture e g physics chemistry games etc computing the perfect and imperfect matching numbers is commonly performed but under different names such as partition function for dimer statistics kekule structures of molecules paving domino problem this paper demonstrate the mathematically beautiful but somewhat mystic relation between the symmetry of a graph and the factorable nature of its perfect matching number there is introduced another interesting relation between the certain series of graphs and a family of orthogonal polynomials through the matching polynomial and topological index that are defined for counting the matching numbers

on the multiplicity of an arbitrary aα eigenvalue of a connected graph abstract the a matrix of a graph g is defined by nikiforov as a g d g 1 a g where 0 1 a g and d g respectively denotes the adjacency matrix and the degree diagonal matrix of g the eigenvalues of a g are called the a eigenvalues of g in this paper we prove that for a connected graph g of order n and with maximum degree 2 and for any 0 1 the multiplicity of an arbitrary a eigenvalue of g which is denoted as m g is bounded by m g 2 n 2 1 the equality holds if and only if g and satisfy one of the following conditions i g k n with n 3 and n 1 ii g c n with even order and 2 2 1 cos 2 j n j 1 2 n 2 2 iii g c n with odd order and 2 2 1 cos 2 j n j 1 2 n 1 2 iv g k n 2 n 2 with n 4 and n 2 applying this result we solve an open conjecture posed by zhou et al in 32 one year ago

skew rank of an oriented graph in terms of matching number abstract an oriented graph g is a digraph without loops and multiple arcs where g is called the underlying graph of g let s g denote the skew adjacency matrix of g the rank of s g is called the skew rank of g denoted by sr g which is even since s g is skew symmetric li and yu 2015 12 proved that the skew rank of an oriented unicyclic graph g is either 2 m g 2 or 2 m g where m g denotes the matching number of g in this paper we extend this result to general cases it is proved that the skew rank of an oriented connected graph g is an even integer satisfying 2 m g 2 g sr g 2 m g where g e g v g 1 is the number of fundamental cycles also called the first betti number besides the oriented graphs satisfying sr g 2 m g 2 g are characterized definitely

on the relationship between the skew rank of an oriented graph and the rank of its underlying graph abstract an oriented graph g is a digraph without loops and multiple arcs where g is the underlying graph of g let s g denote the skew adjacency matrix of g and a g be the adjacency matrix of g the rank resp skew rank of g resp g written as r g resp s r g refers to the rank of a g resp s g it is natural and interesting to study the relationship between s r g and r g wong ma and tian european j combin 54 2016 76 86 30 determined the sharp upper bound on s r g r g as a continuance of it in this paper a sharp lower bound on s r g r g is determined as well a sharp upper bound on s r g r g is determined all the corresponding extremal oriented graphs g are characterized respectively

on the characteristic polynomials and h ranks of the weighted mixed graphs abstract let d g be an n vertex weighted mixed graph with hermitian adjacency matrix h d g the characteristic polynomial of the weighted mixed graph d g is defined as d g det i n h d g r 0 n r n r and the h rank of d g written as r k d g is the rank of h d g in this paper we begin by interpreting all the coefficients of the characteristic polynomial d g then we establish recurrences for the characteristic polynomial d g by these obtained results as a unified approach some main results obtained in electron j combin 18 1 2011 156 linear algebra appl 436 2012 3597 3607 linear algebra appl 466 2015 182 207 can be deduced consequently furthermore for a weighted mixed bipartite cactus graph d g we give a necessary and sufficient condition for r k d g 2 t where t is any integer no greater than the matching number of g finally as an application of our results we study the minimum h rank problem determining the minimum h rank of bipartite cactus graphs

on the nullity and the matching number of unicyclic graphs abstract let g be a graph with n vertices and g be the matching number of g let g denote the nullity of g the multiplicity of the eigenvalue zero of g it is well known that if g is a tree then g n 2 g tan and liu x tan b liu on the nullity of unicyclic graphs linear alg appl 408 2005 212 220 proved that the nullity set of all unicyclic graphs with n vertices is 0 1 n 4 and characterized the unicyclic graphs with g n 4 in this paper we characterize the unicyclic graphs with g n 5 and we prove that if g is a unicyclic graph then g equals n 2 g 1 n 2 g or n 2 g 2 we also give a characterization of these three types of graphs furthermore we determine the unicyclic graphs g with g 0 which answers affirmatively an open problem by tan and liu

the rank of a signed graph in terms of the rank of its underlying graph abstract let g be a signed graph and a be its adjacency matrix where g is the underlying graph of the rank r of is the rank of a we know that for a signed graph g is balanced if and only if g g that is when is balanced then r r g where r g is the rank of the underlying graph g of a natural problem is that how about the relations between the rank of an unbalanced signed graph and the rank of its underlying graph in this paper we first prove that r g 2 d g r r g 2 d g for an unbalanced signed graph with d g 1 where d g e g v g g is the dimension of cycle spaces of g g is the number of connected components of g as an application we also prove that 1 d g r r g 1 d g for an unbalanced signed graph with d g 1 all corresponding extremal graphs are characterized

a characterization of graphs with rank 4 the rank of a graph g is defined to be the rank of its adjacency matrix in this paper we consider the following problem what is the structure of a connected graph with rank 4 this question has not yet been fully answered in the literature and only some partial results are known in this paper we resolve this question by completely characterizing graphs g whose adjacency matrix has rank 4

an artificial intelligence approach for the generation and enumeration of perfect matchings on graphs abstract a new search based algorithm is developed for an effective combinatorial exploration of all the perfect matchings on a general graph the algorithm makes use of a relations based representation of the graph and is completely independent of the nature of the graph though this is a typical np class subgraph enumeration problem the search ranks high in selectivity without compromising accuracy using heuristic guidance since the present algorithm is quite compact nonspecific and generates all perfect matchings of the graph it excels all the earlier algorithms in its uniqueness further it is flexible to accommodate domain specific efficiency improvements which is illustrated for some classes of graphs

regularity results for fully nonlinear parabolic integro differential operators in this paper we consider the regularity theory for fully nonlinear parabolic integro differential equations with symmetric kernels we are able to find parabolic versions of alexandrov backelman pucci estimate with 0 sigma 2 and we show a harnack inequality holder regularity and c 1 alpha regularity of the solutions by obtaining decay estimates of their level sets

kekule structures and their symmetry properties abstract after a short historical introduction the role of kekule structures in theoretical chemistry is summarized the present work concentrates upon conjugated hydrocarbons with six membered rings in a mathematical treatment they are represented by benzenoid systems which consist of congruent regular hexagons the vertices represent carbon atoms relations are given between certain characteristic numbers viz the number of vertices of different kinds secondary tertiary external internal number of edges and of the rings kekulean and non kekulean benzenoids are exemplified the latter class consists of mathematical constructions without any possible kekule structure a great amount of work has been done in the area of graph theory applied to benzenoids some of the most important theorems are summarized another approach employs group theory in the studies of the symmetry of kekule structures a benzenoid or one of its kekule structures may belong to one of the following eight symmetry groups examples are given cata condensed benzenoids are defined by having no internal vertices especially simple are the single straight and single zig zag chains the number of kekule structures and their symmetries are treated in these two cases in the latter case the fibonacci numbers come in some classes of reticular benzenoids are defined and the multiple zig zag chains are considered in some detail a recurrence formula for the number of kekule structures with relevance to this class of benzenoids is presented for the first time along with several deductions from it

on the nullity of bicyclic graphs the nullity of a graph is the multiplicity of the eigenvalue zero in its spectrum in this paper we obtain the nullity set of n vertex unicyclic graphs and characterize the unicyclic graphs with extremal nullity

a short proof of zhou wong and sun s conjecture abstract let g be a connected graph with order n maximum degree 2 and nullity g zhou wong and sun 6 conjectured that g 2 n 2 1 with equality if and only if g is a cycle c n with n divisible by 4 or a complete bipartite graph k n 2 n 2 very recently cheng liu and liu 2 confirmed this conjecture in this paper we give a much shorter proof for this conjecture on the other hand we show that g 2 n 1 if 3 and g k n 2 n 2 k n 1 2 n 1 2 which may deduce a main result obtained in 2

a characterization of long graphs of arbitrary rank abstract the rank of a graph g is defined to be the rank of its adjacency matrix a connected graph g of rank n is called long if it has maximum diameter among all connected graphs with rank n in this paper we characterize the long graphs of rank n for n an arbitrary positive integer

topological aspects of benzenoids and coronoids including snowflakes and laceflowers abstract a coronoid is a benzenoid with a hole some aspects of the studies of topological properties of benzenoids and coronoids are reviewed the survey includes the search for concealed non kekuleans multiple zigzag chains and primitive coronoids enumerations and classifications of polyhexes i e benzenoids and coronoids are treated in some details the original contributions of the present work are concentrated upon polyhexes with hexagonal symmetry benzenoids with hexagonal symmetry are referred to as snowflakes the forms of all snowflakes with d 6 h symmetry and h the number of hexagons 55 are displayed as computer generated drawings in addition the concealed non kekuleans with h 61 and h 67 are shown coronoids with hexagonal symmetry represent another main subject of this work single coronoids in general are classified into regular primitive half essentially disconnected essentially disconnected and non kekulean systems the smallest members of all these categories among coronoids with d 6 h symmetry are displated viz h 36 for the primitive 54 for the non kekulean systems and h 30 for the other categories combinatorial formulas of k the number of kekule structures for some classes of coronoids with hexagonal symmetry are given including a complete solution for the primitive systems belonging to d 6 h or c 6 h finally the forms of the smallest h 49 multiple coronoids with d 6 h symmetry laceflowers are displayed

an analytic approach to purely nonlocal bellman equations arising in models of stochastic control abstract given a bounded domain r d and two integro differential operators l 1 l 2 of the form l j u x p v u x u y k j x y x y d y we study the fully nonlinear bellman equation 0 1 max j 1 2 l j u x a j x u x f j x 0 in with dirichlet boundary conditions here a j f j r are non negative functions we prove the existence of a non negative function u r which satisfies 0 1 almost everywhere the main difficulty arises through the nonlocality of l j and the absence of regularity near the boundary