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Let H be a uniform hypergraph with cut vertices, which is a coalescence of two nontrivial connected sub-hypergraphs (called branches) at some vertex. The traces of the adjacency tensor A (H) of H , simply called the traces of H , are important in the expression of the determinant and the characteristic polynomial of A (H) , and are closely related to the Estrada index of H. In this paper we give a formula for the traces of H in terms of those of its branches, and get some perturbation results on the traces of H when a branch of H attached at one vertex is relocated to another vertex. We prove that among all hypertrees with fixed number of edges, the hyperpath is the unique one with minimum Estrada index and the hyperstar is the unique one with maximum Estrada index. [ABSTRACT FROM AUTHOR]

Let G be a connected m -uniform hypergraph. In this paper we mainly consider the eigenvectors of the Laplacian or signless Laplacian tensor of G associated with zero eigenvalue, called the first Laplacian or signless Laplacian eigenvectors of G. By means of the incidence matrix of G , the number of first Laplacian or signless Laplacian (or H-)eigenvectors can be obtained explicitly by solving the Smith normal form of the incidence matrix over Z m (or Z 2). Consequently, we prove that the number of first Laplacian (H-)eigenvectors is equal to the number of first signless Laplacian (H-)eigenvectors when zero is an (H-)eigenvalue of the signless Laplacian tensor. We establish a connection between first Laplacian (signless Laplacian) H-eigenvectors and the even (odd) bipartitions of G. [ABSTRACT FROM AUTHOR]

Abstract For a nonnegative symmetric weakly irreducible tensor, its spectral radius is an eigenvalue of the tensor corresponding to a unique positive eigenvector called the Perron vector. But including the Perron vector, it may have more than one eigenvector corresponding to the spectral radius. The projective eigenvariety of the tensor associated with the spectral radius is the set of the eigenvectors of the tensor corresponding to the spectral radius considered in the complex projective space. In this paper we proved that such projective eigenvariety admits a module structure, which is determined by the support of the tensor and can be characterized explicitly by the Smith normal form of the incidence matrix of the tensor. We introduced two parameters: the stabilizing index and the stabilizing dimension of the tensor, where the former is exactly the cardinality of the projective eigenvariety and the latter is the composition length of the projective eigenvariety as a module. We give some upper bounds for the two parameters, and characterize the case that there is only one eigenvector of the tensor corresponding to the spectral radius, i.e. the Perron vector. By applying the above results to the adjacency tensor of a connected uniform hypergraph, we give some upper bounds for the two parameters in terms of the structural parameters of the hypergraph such as path cover number, matching number and the maximum length of paths. [ABSTRACT FROM AUTHOR]

Let G be a graph and let A ( G ) be adjacency matrix of G . The positive inertia index (respectively, the negative inertia index) of G , denoted by p ( G ) (respectively, n ( G ) ), is defined to be the number of positive eigenvalues (respectively, negative eigenvalues) of A ( G ) . In this paper, we present the bounds for p ( G ) and n ( G ) as follows: m ( G ) − c ( G ) ≤ p ( G ) ≤ m ( G ) + c ( G ) , m ( G ) − c ( G ) ≤ n ( G ) ≤ m ( G ) + c ( G ) , where m ( G ) and c ( G ) are respectively the matching number and the cyclomatic number of G . Furthermore, we characterize the graphs which attain the upper bounds and the lower bounds respectively. [ABSTRACT FROM AUTHOR]

Let G be a simple graph or hypergraph, and let A ( G ) , L ( G ) , Q ( G ) be the adjacency, Laplacian and signless Laplacian tensors of G respectively. The largest H -eigenvalues (respectively, the spectral radii) of L ( G ) , Q ( G ) are denoted respectively by λ max L ( G ) , λ max Q ( G ) (respectively, ρ L ( G ) , ρ Q ( G ) ). It is known that for a connected non-bipartite simple graph G , λ max L ( G ) = ρ L ( G ) < ρ Q ( G ) . But this does not hold for non-odd-bipartite hypergraphs. We will investigate this problem by considering a class of generalized power hypergraphs G k , k 2 , which are constructed from simple connected graphs G by blowing up each vertex of G into a k 2 -set and preserving the adjacency of vertices. Suppose that G is non-bipartite, or equivalently G k , k 2 is non-odd-bipartite. We get the following spectral properties: (1) ρ L ( G k , k 2 ) = ρ Q ( G k , k 2 ) if and only if k is a multiple of 4; in this case λ max L ( G k , k 2 ) < ρ L ( G k , k 2 ) . (2) If k ≡ 2 ( mod 4 ) , then for sufficiently large k , λ max L ( G k , k 2 ) < ρ L ( G k , k 2 ) . Motivated by the study of hypergraphs G k , k 2 , for a connected non-odd-bipartite hypergraph G , we give a characterization of L ( G ) and Q ( G ) having the same spectra or the spectrum of A ( G ) being symmetric with respect to the origin, that is, L ( G ) and Q ( G ) , or A ( G ) and − A ( G ) are similar via a complex (necessarily non-real) diagonal matrix with modular-1 diagonal entries. So we give an answer to a question raised by Shao et al., that is, for a non-odd-bipartite hypergraph G , that L ( G ) and Q ( G ) have the same spectra can not imply they have the same H -spectra. [ABSTRACT FROM AUTHOR]

The generalized power of a simple graph G , denoted by G k , s , is obtained from G by blowing up each vertex into an s -set and each edge into a k -set, where 1 ≤ s ≤ k 2 . When s < k 2 , G k , s is always odd-bipartite. It is known that G k , k 2 is non-odd-bipartite if and only if G is non-bipartite, and G k , k 2 has the same adjacency (respectively, signless Laplacian) spectral radius as G . In this paper, we prove that, regardless of multiplicities, the H -spectrum of A ( G k , k 2 ) (respectively, Q ( G k , k 2 ) ) consists of all eigenvalues of the adjacency matrices (respectively, the signless Laplacian matrices) of the connected induced subgraphs (respectively, modified induced subgraphs) of G . As a corollary, G k , k 2 has the same least adjacency (respectively, least signless Laplacian) H -eigenvalue as G . We also discuss the limit points of the least adjacency H -eigenvalues of hypergraphs, and construct a sequence of non-odd-bipartite hypergraphs whose least adjacency H -eigenvalues converge to − 2 + 5 . [ABSTRACT FROM AUTHOR]

In order to investigate the non-odd-bipartiteness of even uniform hypergraphs, starting from a simple graph G , we construct a generalized power of G , denoted by G k , s , which is obtained from G by blowing up each vertex into a s -set and each edge into a ( k − 2 s ) -set, where s ≤ k / 2 . When s < k / 2 , G k , s is always odd-bipartite. We show that G k , k 2 is non-odd-bipartite if and only if G is non-bipartite, and find that G k , k 2 has the same adjacency (respectively, signless Laplacian) spectral radius as G . So the results involving the adjacency or signless Laplacian spectral radius of a simple graph G hold for G k , k 2 . In particular, we characterize the unique graph with minimum adjacency or signless Laplacian spectral radius among all non-odd-bipartite hypergraphs G k , k 2 of fixed order, and prove that 2 + 5 is the smallest limit point of the non-odd-bipartite hypergraphs G k , k 2 . In addition we obtain some results for the spectral radii of the weakly irreducible nonnegative tensors. [ABSTRACT FROM AUTHOR]

Abstract: Let G be a graph and let be the adjacency matrix of G. The signature of G is the difference between the positive inertia index and the negative inertia index of . Ma et al. (2013) [10] conjectured that , where and respectively denote the number of cycles in G which have length and for some integers , and proved the conjecture holds for trees, unicyclic or bicyclic graphs. It is known that if G is bipartite, and the signature is closely related to the odd cycles or nonbipartiteness of a graph from the existing results. In this paper we show that the conjecture holds for line graphs and power trees. [Copyright &y& Elsevier]

Abstract: In this paper, we determine the graph whose least eigenvalue of the signless Laplacian attains the minimum or maximum among all connected non-bipartite graphs of fixed order and given number of pendant vertices. Thus we obtain a lower bound and an upper bound for the least eigenvalue of the signless Laplacian of a graph in terms of the number of pendant vertices. [Copyright &y& Elsevier]

Abstract: In this paper we introduce the nullity of signed graphs, and give some results on the nullity of signed graphs with pendant trees. We characterize the unicyclic signed graphs of order n with nullity respectively. [Copyright &y& Elsevier]

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 pendent edges on each vertex in the cycle (for some integer ), denoted by . In this paper, we give a complete characterization on the nullity of the line graph as follows: Let . Then [(i)] if and only if and . [(ii)] if and only if and either [(1)] for , the length of any zero chain of is even and ; or [(2)] and one of and must hold. [(iii)] , otherwise. [Copyright &y& Elsevier]

Abstract: Let G be a simple graph, and let the least eigenvalue of the signless Laplacian of the graph G. In this paper we focus on the relations between the least eigenvalue and some parameters reflecting the graph bipartiteness. We introduce two parameters: the vertex bipartiteness and the edge bipartiteness , and show thatWe also define another parameter involved with a cut set, and prove thatwhere is the maximum degree of the graph G. The above two inequalities are very similar in form to those given by Fiedler and Mohar, respectively, with respect to the algebraic connectivity of Laplacian of graphs, which is used to characterize the connectedness of graphs. [Copyright &y& Elsevier]

Abstract: We investigate how the least eigenvalue of the signless Laplacian of a graph changes by relocating a bipartite branch from one vertex to another vertex, and minimize the least eigenvalue of the signless Laplacian among the class of connected graphs with fixed order which contains a given non-bipartite graph as an induced subgraph. [Copyright &y& Elsevier]

Abstract: Let be the set of the complements of trees of order n. In this paper, we characterize the unique graph whose least eigenvalue attains the minimum among all graphs in . [Copyright &y& Elsevier]

Abstract: The nullity of a graph is defined as the multiplicity of the eigenvalue zero in the spectrum of the adjacency matrix of the graph. We investigate a class of graphs with pendant trees, and express the nullity of such graph in terms of that of its subgraphs. As an application of our results, we characterize unicyclic graphs with a given nullity. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we characterize the graphs with maximum signless Laplacian or adjacency spectral radius among all graphs with fixed order and given vertex or edge connectivity. We also discuss the minimum signless Laplacian or adjacency spectral radius of graphs subject to fixed connectivity. Consequently we give an upper bound of signless Laplacian or adjacency spectral radius of graphs in terms of connectivity. In addition we confirm a conjecture of Aouchiche and Hansen involving adjacency spectral radius and connectivity. [Copyright &y& Elsevier]

Abstract: In this paper we characterize the unique graph whose least eigenvalue attains the minimum among all graphs of a fixed order and a given vertex (edge) independence number or vertex (edge) cover number, and get some bounds for the vertex (edge) independence number, vertex (edge) cover number of a graph in terms of the least eigenvalue of the graph. [Copyright &y& Elsevier]

Abstract: In this paper we characterize the unique graph whose least eigenvalue attains the minimum among all connected graphs of fixed order and given number of cut vertices, and then obtain a lower bound for the least eigenvalue of a connected graph in terms of the number of cut vertices. During the discussion we also get some results for the spectral radius of a connected bipartite graph and its upper bound. [Copyright &y& Elsevier]

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 , and characterize the bipartite graphs with nullity and the regular bipartite graphs with nullity . [Copyright &y& Elsevier]

Abstract: Let be a simple graph and be the adjacency matrix of . The eigenvalues of are those of . In this paper, we characterize the graphs with the minimal least eigenvalue among all graphs of fixed order with given vertex connectivity or edge connectivity. [Copyright &y& Elsevier]

Abstract: A connected graph is said to be unoriented Laplacian maximizing if the spectral radius of its unoriented Laplacian matrix attains the maximum among all connected graphs with the same number of vertices and the same number of edges. A graph is said to be threshold (maximal) if its degree sequence is not majorized by the degree sequence of any other graph (and, in addition, the graph is connected). It is proved that an unoriented Laplacian maximizing graph is maximal and also that there are precisely two unoriented Laplacian maximizing graphs of a given order and with nullity 3. Our treatment depends on the following known characterization: a graph is threshold (maximal) if and only if for every pair of vertices of , the sets , where denotes the neighbor set of in , are comparable with respect to the inclusion relation (and, in addition, the graph is connected). A conjecture about graphs that maximize the unoriented Laplacian matrix among all graphs with the same number of vertices and the same number of edges is also posed. [Copyright &y& Elsevier]

Abstract: 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 denote the set of connected unicyclic graphs of order n and girth k, and let 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 . We, finally, characterize the unique graph with minimum least eigenvalue (respectively, the unique graph with maximum spread) among all graphs in . [Copyright &y& Elsevier]

Abstract: Let T be a tree and L be its Laplacian matrix. With respect to a given vector Y which gives a valuation of vertices of T, a vertex u of T is called a characteristic vertex if Y[u]=0 and if there is a vertex w adjacent to u with Y[w]≠0; and an edge {u, w} of T is called a characteristic edge if Y[u]Y[w]<0. The characteristic set of T with respect to Y, denoted by , is defined as the collection of all characteristic vertices and characteristic edges of T. For the kth smallest eigenvalue λ k (k ⩾2) of T, if λk > λ k−1, the corresponding eigenvector Y of λ k is called a k-vector. A tree T is called k-simple if for all k-vectors Y. We show that k-simple trees exist and characterize them. We also show the characteristic sets determined by all the k-vectors is the same, which is consistent with the property of 2-simple tree (i.e., arbitrary tree). Finally, we give some properties of the eigenvalues and eigenvectors of a k-simple tree. [Copyright &y& Elsevier]

Abstract: Let G =(V, E) be a simple graph and {λ 1(G),…, λ n (G)} be its adjacency spectrum. It is easy to see that if an edge is added between two isolated vertices, then one zero eigenvalue increases by 1, and another zero eigenvalue decreases by 1. Let G + be a connected graph obtained from G by adding an edge e ∉ E(G). In this paper, it will be proved that the spectrum of G + is different from that of G only in two places with one eigenvalue increases by m and another eigenvalue decreases by m, where m >0 is a rational number, if and only if G is an empty graph with order 2. It will also be proved that one cannot construct a new adjacency integral connected graph with order n ⩾3 from a known one by adding an edge. [Copyright &y& Elsevier]

In this paper, we characterize the mixed graphs with exactly one Laplacian eigenvalue moving up by an integer and other Laplacian eigenvalues remaining invariant when an edge is added. The results extend those of Fan [Linear and Multilinear Algebra 50 (2002) 133] for general graphs, and So [Linear and Multilinear Algebra 46 (1999) 193] for simple graphs. [Copyright &y& Elsevier]

Let G be a t -uniform hypergraph, and let c (G) denote the cyclic index of the adjacency tensor of G. Let m , s be positive integers such that s ≥ 2 and m = s t. The generalized power G m , s of G is obtained from G by blowing up each vertex into an s -set and preserving the adjacency relation. It was conjectured that c (G m , s) = s ⋅ c (G). In this paper, by using a matrix equation over Z m that characterizes the spectral symmetry or cyclic index of an m -uniform hypergraph, we give an equivalent condition for the equality in the conjecture. Finally we show that the conjecture is false by a counterexample. [ABSTRACT FROM AUTHOR]

Let G be a connected non-odd-bipartite hypergraph with even uniformity. The least H-eigenvalue of the signless Laplacian tensor of G is simply called the least eigenvalue of G and the corresponding H-eigenvectors are called the first eigenvectors of G. In this paper we give some numerical and structural properties about the first eigenvectors of G which contains an odd-bipartite branch, and investigate how the least eigenvalue of G changes when an odd-bipartite branch attached at one vertex is relocated to another vertex. We characterize the hypergraph(s) whose least eigenvalue attains the minimum among a certain class of hypergraphs which contain a fixed non-odd-bipartite connected hypergraph. Finally we present some upper bounds of the least eigenvalue and prove that zero is the least limit point of the least eigenvalues of connected non-odd-bipartite hypergraphs. [ABSTRACT FROM AUTHOR]

Let G be a connected graph and let T be an acyclic set of edges of G. A partial orientation σ of G with respect to T is an orientation of the edges of G except those edges of T , the resulting graph associated with which is denoted by G T σ. In this paper we prove that there exists a partial orientation σ of G with respect to T such that the largest eigenvalue of the Hermitian adjacency matrix of G T σ is at most the largest absolute value of the roots of the matching polynomial of G. [ABSTRACT FROM AUTHOR]

Let G be a complex unit gain graph which is obtained from an undirected graph Γ by assigning a complex unit φ ( v i v j ) to each oriented edge v i v j such that φ ( v i v j ) φ ( v j v i ) = 1 for all edges. The Laplacian matrix of G is defined as L ( G ) = D ( G ) − A ( G ) , where D ( G ) is the degree diagonal matrix of Γ and A ( G ) = ( a i j ) has a i j = φ ( v i v j ) if v i is adjacent to v j and a i j = 0 otherwise. In this paper, we provide a combinatorial description of det ( L ( G ) ) that generalizes that for the determinant of the Laplacian matrix of a signed graph. [ABSTRACT FROM AUTHOR]

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. In a recent work the trees with maximal Laplacian spread and with minimal Laplacian spread among all trees of fixed order are separately determined. In this work, we characterize the unique unicyclic graph with maximal Laplacian spread among all connected unicyclic graphs of fixed order. [Copyright &y& Elsevier]

Abstract: In this paper we introduce a new parameter for a graph called the minimum universal rank. This parameter is similar to the minimum rank of a graph. For a graph G the minimum universal rank of G is the minimum rank over all matrices of the form where A is the adjacency matrix of G, J is the all ones matrix and D is the matrix with the degrees of the vertices in the main diagonal, and are scalars. Bounds for general graphs based on known graph parameters are given, as is a formula for the minimum universal rank for regular graphs based on the multiplicity of the eigenvalues of A. The exact value of the minimum universal rank of some families of graphs are determined, including complete graphs, complete bipartite graph, paths and cycles. Bounds on the minimum universal rank of a graph obtained by deleting a single vertex are established. It is shown that the minimum universal rank is not monotone on induced subgraphs, but bounds based on certain induced subgraphs, including bounds on the union of two graphs, are given. [Copyright &y& Elsevier]

Abstract: A sufficient condition of adaptive synchronization in dynamical networks with coupling delays is gained, where the coupling configuration corresponds to a weighted graph. One can reduce edges to original graph such that the network more quickly achieves synchronization. Numerical simulations are given to illustrate the efficiency of theoretical results. [Copyright &y& Elsevier]