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In this paper, we investigate the traces of the adjacency tensor of uniform hypergraphs (simply called the traces of hypergraphs). We give new expressions for the traces of hypertrees and linear unicyclic hypergraphs by the weight function assigned to their connected sub-hypergraphs, and present some perturbation results for the traces of a hypergraph when its structure is locally changed. As an application we determine the unique hypertree with maximum Estrada index among all hypertrees with perfect matchings and given number of edges, and the unique unicyclic hypergraph with maximum Estrada index among all unicyclic hypergraph with girth 3 and given number of edges. [ABSTRACT FROM AUTHOR]

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]

A hypergraph is said to be oriented if each edge-vertex incidence has a label of +1 or −1. An oriented hypergraph is called incidence balanced if there exists a bipartition of the vertex set such that every edge intersects one part of the bipartition in positively incident vertices with the edge and the other part in negatively incident vertices with the edge. In this paper, we investigate the incidence balance of oriented hypergraphs and induced signed hypergraphs by means of the eigenvalues of associated matrices and tensors and provide a spectral method to characterize the incidence balanced oriented hypergraphs and their induced signed hypergraphs. [ABSTRACT FROM AUTHOR]

Let H be a connected m -uniform hypergraph, and let A (H) be the adjacency tensor of H whose spectrum is simply called the spectrum of H. Let s (H) denote the number of eigenvectors of A (H) associated with the spectral radius, and c (H) denote the number of eigenvalues of A (H) with modulus equal to the spectral radius, which are respectively called the stabilizing index and cyclic index of H. Let H ¯ be a k -fold covering of H which can be obtained from some permutation assignment in the symmetric group S k on H. In this paper, we first characterize the connectedness of H ¯ by its incidence graph and the permutation assignment, and then investigate the relationship between the spectral property of H and that of H ¯. By applying module theory and group representation, if H ¯ is connected, we prove that s (H) | s (H ¯) and c (H) | c (H ¯). In particular, when H ¯ is a 2-fold covering of H , if m is even, we show that regardless of multiplicities, the spectrum of H ¯ contains the spectrum of H and the spectrum of a signed hypergraph with H as underlying hypergraph; if m is odd, we give an explicit formula for s (H ¯). We also find some differences on the spectral property between hypergraph coverings and graph coverings by examples. [ABSTRACT FROM AUTHOR]

The spectral radius of a uniform hypergraph is defined to be that of the adjacency tensor of the hypergraph. It is known that the unique unicyclic hypergraph with the largest spectral radius is a nonlinear hypergraph, and the unique linear unicyclic hypergraph with the largest spectral radius is a power hypergraph. In this paper we determine the unique linear unicyclic hypergraph with the second or third largest spectral radius, where the former hypergraph is a power hypergraph and the latter hypergraph is a non-power hypergraph. [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 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]

Let G be connected uniform hypergraph and let A (G) be the adjacency tensor of G. The stabilizing index of G is exactly the number of eigenvectors of A (G) associated with the spectral radius, and the cyclic index of G is exactly the number of eigenvalues of A (G) with modulus equal to the spectral radius. Let G 1 ⊙ G 2 and G 1 □ G 2 be the coalescence and Cartesian product of connected m -uniform hypergraphs G 1 and G 2 respectively. In this paper, we give explicit formulas for the stabilizing indices and cyclic indices of G 1 ⊙ G 2 and G 1 □ G 2 in terms of those of G 1 and G 2 or the invariant divisors of their incidence matrices over Z m , respectively. [ABSTRACT FROM AUTHOR]

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]