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In this paper we investigate the hypergraphs whose spectral radii attain the maximum among all uniform hypergraphs with given number of edges. In particular we characterize the hypergraph(s) with maximum spectral radius over all unicyclic hypergraphs, linear or power unicyclic hypergraphs with given girth, linear or power bicyclic hypergraphs, respectively.

For a nonnegative weakly irreducible tensor, its spectral radius is an eigenvalue corresponding to a unique positive eigenvector up to a scalar, called Perron vector. But including the Perron vector, it may have more than one eigenvector corresponding to the spectral radius. The projective eigenvariety associated with the spectral radius is the set of the eigenvectors corresponding to the spectral radius considered in the complex projective space. We prove that the dimension of the above projective eigenvariety is zero, i.e. there are finitely many eigenvectors associated with the spectral radius up to a scalar. We characterize a nonnegative combinatorially symmetric tensor for which the dimension of projective eigenvariety associated with spectral radius is greater than zero. Finally we apply those results to the adjacency tensors of uniform hypergraphs. [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]

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: 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 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]

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]

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]

The edge blow-up of a graph G , denoted by G p + 1 , is obtained by replacing each edge of G with a clique of order p + 1 , where the new vertices of the cliques are all distinct. Yuan (2022) [20] determined the range of the Turán numbers for edge blow-up of all bipartite graphs and the exact Turán numbers for edge blow-up of all non-bipartite graphs. In this paper, we prove that the graphs with the maximum spectral radius in an n -vertex graph without any copy of edge blow-up of star forests are the extremal graphs for edge blow-up of star forests when n is sufficiently large. [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]