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 k-set and each edge into a (k − 2s)-set, where2 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 nonodd-bipartite hypergraphs G k, k 2 . In addition we obtain some results for the spectral radii of the weakly irreducible nonnegative tensors.

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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

In this paper we establish some spectral conditions for a graph to be Hamilton-connected in terms of the spectral radius of the adjacency matrix or the signless Laplacian of the graph or its complement. For the existence of Hamiltonian paths or cycles in a graph, we also give a sufficient condition by the signless Laplacian spectral radius.

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 n − 2, n − 3, n − 4, n − 5 respectively.

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Let A be a weakly irreducible nonnegative tensor with spectral radius ρ(A). Let D (respectively, D (0) ) be the set of normalized diagonal matrices arising from the eigenvectors of A corresponding to the eigenvalues with modulus ρ(A) (respectively, the eigenvalue ρ(A)). It is shown that D is an abelian group containing D (0) as a subgroup, which acts transitively on theThe spectral symmetry of A is characterized by the group D/D (0) , and A is called spectral ℓ-symmetric. We obtain the structural information of A by analyzing the property of D, especially for connected hypergraphs we get some results on the edge distribution and coloring. If moreover A is symmetric, we prove that A is spectral ℓ-symmetric if and only if it is (m, ℓ)-colorable. We characterize the spectral ℓ-symmetry of a tensor by using its generalized traces, and show that for an arbitrarily given integer m ≥ 3 and each positive integer ℓ with ℓ | m, there always exists an m-uniform hypergraph G such that G is spectral ℓ-symmetric.2000 Mathematics Subject Classification. Primary 15A18, 05C65; Secondary 13P15, 05C15.

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