hermitian adjacency spectrum and switching equivalence of mixed graphs abstract it is shown that an undirected graph g is cospectral with the hermitian adjacency matrix of a mixed graph d obtained from a subgraph h of g by orienting some of its edges if and only if h g and d is obtained from g by a four way switching operation if g is connected this happens if and only if 1 g 1 d all mixed graphs of rank 2 are determined and this is used to classify which mixed graphs of rank 2 are cospectral with respect to their hermitian adjacency matrix several families of mixed graphs are found that are determined by their hermitian spectrum in the sense that they are cospectral precisely to those mixed graphs that are switching equivalent to them

bicyclic graphs with small positive index of inertia abstract the positive index of inertia of a graph g denoted by i g is the number of the positive eigenvalues of the adjacency matrix of g in this paper we investigate the minimal positive index of inertia among all bicyclic graphs of order n with pendant vertices and characterize the bicyclic graphs with positive index 1 or 2 among all bicyclic graphs of order n

analyzing lattice networks through substructures abstract analyzing the topology of network structures is an important topic studied from many different aspects of science and mathematics the wiener polarity index number of unordered pairs of vertices at distance 3 from each other is one of the representative descriptors of graph structures it was computed for several lattice networks by chen et al 11 in an effort to understand the properties of these networks the wiener polarity index is a variation of the classic distance based graph invariant the wiener index sum of distances between all pairs of vertices which is known to be closely related to the number of substructures in this paper we examine the numbers of various subgraphs of order 4 for these lattice networks in addition to confirming their symmetric nature comparing the numbers of various substructures leads to insights on other less trivial characteristics of these network structures of common interest

hermitian laplacian matrix and positive of mixed graphs a mixed graph is obtained from an undirected graph by orienting a subset of its edges the hermitian adjacency matrix of a mixed graph m of order n is the n n matrix h m h k l where h k l h l k i i 1 if there exists an orientation from vk to vl and h k l h l k 1 if there exists an edge between vk and vl but not exist any orientation and h k l 0 otherwise the value of a mixed walk w v 1 v 2 v 3 v l is h w h 12 h 23 h l 1 l a mixed walk is positive negative if h w 1 h w 1 a mixed cycle is called positive if its value is 1 a mixed graph is positive if each of its mixed cycle is positive in this work we firstly present the necessary and sufficient conditions for the positive of a mixed graph secondly we introduce the incident matrix and hermitian laplacian matrix of a mixed graph and derive some results about the hermitian laplacian spectrum

structure connectivity and substructure connectivity of k ary n cube networks abstract the k ary n cube is one of the most attractive interconnection networks for parallel and distributed computing system in this paper we investigate the fault tolerant capabilities of k ary n cubes with respect to the structure connectivity and substructure connectivity let h is a connected graph the structure connectivity of a graph g denoted by g h is the minimum cardinality of a set of connected subgraphs in g whose deletion disconnects the graph g and every element in the set is isomorphic to h the substructure connectivity of a graph g denoted by s g h is the minimum cardinality of a set of connected subgraphs in g whose deletion disconnects the graph g and every element in the set is isomorphic to a connected subgraph of h we show q n k h and s q n k h for each h k1 k1 1 k1 2 k1 3

generalized measures of fault tolerance for bubble sort networks abstract the k and k are two generalized measurements for fault tolerance of large scale processing system for the bubble sort networks bn this paper determines k b n k b n 2 k n k 1 for k n 2 the results show that to disconnect bn with each vertex in resulting graph has at least k fault free neighbors at least 2 k n k 1 faulty vertices or faulty edges have to occur in particular the results also settle affirmatively a conjecture proposed by shi and wu acta math appl sin e 33 4 2017 933 944

extremality of degree based graph entropies abstract many graph invariants have been used for the construction of entropy based measures to characterize the structure of complex networks based on shannon s entropy we study graph entropies which are based on vertex degrees by using so called information functionals when considering shannon entropy based graph measures there has been very little work to find their extremal values the main contribution of this paper is to prove some extremal values for the underlying graph entropy of certain families of graphs and to find the connection between the graph entropy and the sum of degree powers further conjectures to determine extremal values of graph entropies are given

inequalities and identities for generalized matrix functions abstract let be a character on the symmetric group s n and let a a ij be an n by n matrix the function d a s n n t 1 a t t is called a generalized matrix function if is an irreducible character then d is called an immanent for example if is the alternating character then d is the determinant and if 1 then d is called the permanent denoted per suppose that a is positive semidefinite hermitian we prove that the inequality 1 id d a per a holds for a variety of characters including the irreducible ones corresponding to the partitions n 1 1 and n 2 1 1 of n the main technique used to prove these inequalities is to express the immanents as sums of products of principal subpermanents these expressions for the immanents come from analogous expressions for schur polynomials by means of a correspondence of d e littlewood

conditional connectivity of recursive interconnection networks respect to embedding restriction abstract large scale multiprocessor systems always take some recursive interconnection networks as underlying topologies let g n be an n dimensional recursive interconnection network the m embedding restricted connectivity m g n resp the m embedding restricted edge connectivity m g n of g n is the cardinality of a minimum subset of nodes resp edges if any whose deletion disconnects g n and each node of the remaining components lies in an undamaged m dimensional sub network g m in this paper we present some relationships between the proposed indices and other conditional connectivity indices in general recursive interconnection networks we give some bounds on these two indices in k ary n cubes and bubble sort networks in addition we determine these two indices in k ary n cubes and bubble sort networks in some cases

on weighted directed graphs the study of a mixed graph and its laplacian matrix have gained quite a bit of interest among the researchers mixed graphs are very important for the study of graph theory as they provide a setup where one can have directed and undirected edges in the graph in this article we present a more general structure namely the weighted directed graphs and supply appropriate generalizations of several existing results for mixed graphs related to singularity of the corresponding laplacian matrix we also prove many new combinatorial results relating the laplacian matrix and the graph structure

bounds for degree based network entropies in this paper we continue studying degree based entropies for networks the quantities represent information theoretic network measures which are based on using information functionals involving vertex degrees we prove bounds for entropies which are based on information functionals using degree powers and come up with interrelations between different measures such interrelations are important to study connections between the measures required to understand the measures in depth

embedded connectivity of recursive networks let gn be an n dimensional recursive network the h embedded connectivity h gn resp edge connectivityh gn of gn is the minimum number of vertices resp edges whose removal results in disconnected and each vertex is contained in an h dimensional subnetwork gh this paper determinesh andh for the hypercube qn and the star graph sn and 3 for the bubble sort network bn

complete solution to a problem on the maximal energy of unicyclic bipartite graphs abstract the energy of a simple graph g denoted by e g is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix denote by c n the cycle and p n 6 the unicyclic graph obtained by connecting a vertex of c 6 with a leaf of p n 6 caporossi et al conjectured that the unicyclic graph with maximal energy is p n 6 for n 8 12 14 and n 16 in hou et al 2002 y hou i gutman c woo unicyclic graphs with maximal energy linear algebra appl 356 2002 27 36 the authors proved that e p n 6 is maximal within the class of the unicyclic bipartite n vertex graphs differing from c n and they also claimed that the energies of c n and p n 6 is quasi order incomparable and left this as an open problem in this paper by utilizing the coulson integral formula and some knowledge of real analysis especially by employing certain combinatorial techniques we show that the energy of p n 6 is greater than that of c n for n 8 12 14 and n 16 which completely solves this open problem and partially solves the above conjecture

on the laplacian coefficients of acyclic graphs abstract let g be a graph of order n and let g k 0 n 1 k c k n k be the characteristic polynomial of its laplacian matrix zhou and gutman recently proved that among all trees of order n the kth coefficient ck is largest when the tree is a path and is smallest for stars a new proof and a strengthening of this result is provided a relation to the wiener index is discussed

the le varieties ii for a complex polynomial f n 1 0 0 with a singular set of complex dimensions at the origin we define a sequence of varieties the le varieties off at 0 the multiplicities of these varieties generalize the milnor number for an isolated singularity in particular we show that ifs n 2 the milnor fibre off is obtained fromb 2n by successively attaching k handles wheren s k n ifs n 1 the milnor fibre off is obtained from a2n manifold with the homotopy type of a bouquet of circles by successively attaching k handles where 2 k n

hermitian normalized laplacian matrix for directed networks abstract in this paper we extend and generalize the spectral theory of undirected networks towards directed networks by introducing the hermitian normalized laplacian matrix for directed networks in order to start we discuss the courant fischer theorem for the eigenvalues of hermitian normalized laplacian matrix based on the courant fischer theorem we obtain a similar result towards the normalized laplacian matrix of undirected networks for each i 1 2 n any eigenvalue of hermitian normalized laplacian matrix i 0 2 moreover we prove some special conditions if 0 or 2 is an eigenvalue of the hermitian normalized laplacian matrix l x on top of that we investigate the symmetry of the eigenvalues of l x and the edge version for the eigenvalue interlacing result finally we present two expressions for the coefficients of the characteristic polynomial of the hermitian normalized laplacian matrix as an outlook we sketch some novel and intriguing problems to which our apparatus could generally be applied

restricted connectivity for three families of interconnection networks vertex connectivity and edge connectivity are two important parameters in interconnection networks even though they reflect the fault tolerance correctly they undervalue the resilience of large networks by the concept of conditional connectivity and super connectivity the concept of restricted vertex connectivity and restricted edge connectivity of graphs was proposed by esfahanian a h esfahanian generalized measures of fault tolerance with application to n cube networks ieee transactions on computers 38 1989 1586 1591 such measures take the resilience of large networks into consideration in this paper we propose three families of interconnection networks and discuss their restricted vertex connectivity and restricted edge connectivity in particular the hypercubes twisted cubes crossed cubes mobius cubes star graphs pancake graphs recursive circulant graphs and k ary n cubes are special cases of these families

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

3 regular mixed graphs with optimum hermitian energy abstract let g be a simple undirected graph and g be a mixed graph of g with the generalized orientation and hermitian adjacency matrix h g then g is called the underlying graph of g the hermitian energy of the mixed graph g denoted by e h g is defined as the sum of all the singular values of h g a k regular mixed graph on n vertices having hermitian energy n k is called a k regular optimum hermitian energy mixed graph liu and li 2015 11 proposed the problem of determining all the k regular connected optimum hermitian energy mixed graphs this paper is to give a solution to the problem for the case k 3

hamiltonian laceability of bubble sort graphs with edge faults it is known that the n dimensional bubble sort graph b n is bipartite n 1 regular and has n vertices we first show that for any vertex v b n v has a hamiltonian path between any two vertices in the same partite set without v let f be a subset of edges of b n we next show that b n f has a hamiltonian path between any two vertices of different partite sets if f is at most n 3 then we also prove that b n f has a path of length n 2 between any pair of vertices in the same partite set

a kind of conditional connectivity of cayley graphs generated by unicyclic graphs abstract for a graph g v e a subset f v g is called an r k vertex cut of g if g f is disconnected and each vertex u v g f has at least k neighbors in g f the r k vertex connectivity of g denoted by k g is the cardinality of the minimum r k vertex cut of g which is a refined measure for the fault tolerance of network g in this paper we study 2 for cayley graphs generated by unicyclic graphs let sym n be the symmetric group on 1 2 n and t be a set of transpositions of sym n let g t be the graph on vertex set 1 2 n and edge set ij ij t if g t is a unicyclic graph then simply denote the cayley graph cay sym n t by ug n in particular if g t is a cycle then ug n is the well known modified bubble sort graph mb n we prove in this paper that if m is the length of the unique cycle in g t then 2 ug n 4 n 10 if 3 m n 4 n 8 if 4 m n

a kind of conditional vertex connectivity of cayley graphs generated by 2 trees let g be a graph then t v g is called an r k vertex cut if g t is disconnected and each vertex in v g t has at least k neighbors in g t the size of a smallest r k vertex cut is the r k vertex connectivity of g and is denoted by k k g in this paper we determine the numbers k 1 and k 2 for cayley graphs generated by 2 trees including the popular alternating group graphs

fault tolerance in bubble sort graph networks the bubble sort graph b n is one of the attractive underlying topologies for distributed systems f b n k resp f b n k is the minimum number of faulty nodes resp links that make every n k dimensional sub bubble sort graph faulty in b n under node resp link failure model in this paper we proved that f b n 0 f b n 0 1 f b n 1 f b n 1 n for n 4 f b n 2 f b n 2 n 2n 3 for n 6 f b n n 2 n 2 for n 3 f b n n 2 n 1 n 2 for n 3 f b n n 1 n for n 2 and f b n k f b n k k 1 2 k c k n for n 6 and 3 k n 3

hook immanantal inequalities for trees explained abstract let d k denote the normalized hook immanant corresponding to the partition k ln k of n p heyfron proved the family of immanantal inequalities det a d 1 a d 2 a d n a per a 1 for all positive semidefinite hermitian matrices a motivated by a conjecture of r merris it was shown by the authors that 1 may be improved to d k 1 l t k 2 k 1 d k l t 2 for all 2 k n whenever l t is the laplacian matrix of a tree t the proof of 2 relied on rather involved recursive relations for weighted matchings in the tree t as well as identities of hook characters in this work we circumvent this tedium with a new proof using the notion of vertex orientations this approach makes 2 immediately apparent and more importantly provides an insight into why it holds namely the absence of certain vertex orientations for all trees as a by product we obtain an improved bound 0 1 k 1 d k l t d k l s n k 2 k 1 d k l t d k 1 l t where s n is the star with n vertices the ease with which the inequality in 2 and its improvement are derived points to the value of the concept of vertex orientation in the study of immanantal inequalities on graphs

k restricted edge connectivity for some interconnection networks abstract k restricted edge connectivity is an important parameter in measuring the reliability and fault tolerance of large interconnection networks in this paper we present two families of graphs similar with the networks proposed by chen et al y c chen j j m tan l h hsu s s kao super connectivity and super edge connectivity for some interconnection networks applied mathematics and computation 140 2003 245 254 and study the k k 2 3 restricted edge connectivity of these graphs in particular as the applications of our results the k k 2 3 restricted edge connectivity of the recursive circulant graphs and the n ary k cubes is given

extraconnectivity of k ary n cube networks given a graph g and a non negative integer g the g extraconnectivity of g is the minimum cardinality of a set of vertices in g if such a set exists whose deletion disconnects g and leaves every remaining component with more than g vertices this study shows that the 2 extraconnectivity of a k ary n cube q n k for k 4 and n 5 is equal to 6n 5

hermitian adjacency matrices and hermitian energies of mixed graphs abstract a complex adjacency matrix of a mixed graph is introduced in the present paper which is a hermitian matrix and called the hermitian adjacency matrix it incorporates both adjacency matrix of an undirected graph and skew adjacency matrix of an oriented graph some of its properties are studied furthermore properties of its characteristic polynomial are studied cospectral problems among mixed graphs including mixed graphs and their underlying graphs oriented graphs and their underlying graphs are studied we give equivalent conditions for a mixed graph especially oriented graph that share the same spectrum with its underlying graph as a consequence we reconfirm a conjecture which was proposed by cui and hou in ref 8 we also show that the spectrum of the hermitian matrix of a mixed graph is invariant when changing the value of any its cut edge if any correspondingly an energy of a mixed graph is introduced and called the hermitian energy it incorporates both the energy of an undirected graph and the skew energy of an oriented graph some of its bounds are given especially the mixed graphs with optimal upper bound of hermitian energy are characterized an infinite family of mixed graphs attaining the maximum hermitian energy is constructed moreover the hermitian energy of a mixed tree is showed to be equal to the energy of its underlying tree finally the integral formula for hermitian energy of a mixed graph is given

principal minor version of matrix tree theorem for mixed graphs in yu etal 2017 an analytical expression of the determinant of the hermitian quasi laplacian matrix of mixed graphs has been proven in this paper we are going to extend those results and derive an analytical expression for the principal minors of the hermitian quasi laplacian matrix which is the principal minor version of the matrix tree theorem

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

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

conditional connectivity of star graph networks under embedding restriction the n dimensional star graph s n is one of the most attractive interconnection networks of large scale multiprocessor systems the k embedding restricted edge connectivity h k s n z k s n of s n is defined to be the cardinality of a minimum subset of edges nodes if any whose deletion disconnects s n and each node of the remaining components lies in an undamaged k dimensional substar s k in this paper we investigate the k embedding restricted edge connectivity of s n and determine the values of h k s n and z k s n for some k s

a q analogue of the distance matrix of a tree we consider a q analogue of the distance matrix called the q distance matrix of an unweighted tree and give formulae for the inverse and the determinant which generalize the existing formulae for the distance matrix we obtain the smith normal form of the q distance matrix of a tree the relationship of the q distance matrix with the laplacian matrix leads to q analogue of the laplacian matrix of a tree some of whose properties are also given we study another matrix related to the distance matrix the exponential distance matrix and show its relationship with the q laplacian and the q distance matrix a formula for the determinant of the q distance matrix of a weighted tree is also given

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

note on two generalizations of the randic index for a given graph g the well known randic index of g introduced by milan randic in 1975 is defined as r g u v e g d u d v 1 2 where the sum is taken over all edges uv and du denotes the degrees of u bollobas and erdos generalized this index by replacing 1 2 with any real number which is called the general randic index dvo ak et al introduced a modified version of randic index r g u v e g max d u d v 1 based on this recently knor et al introduced two generalizations r g u v e g min d u d v and r g u v e g max d u d v for any real number clearly the former is a lower bound for the general randic index and the latter is its upper bound knor et al studied extremal values of r g and r g and concluded some open problems in this paper we consider the open problems and give some comments and results some results for chemical trees are obtained