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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]

A graph in a certain graph class is called minimizing if the least eigenvalue of its adjacency matrix attains the minimum among all graphs in that class. Bell et al. have identified a subclass within the connected graphs of order n and size m in which minimizing graphs belong (the complements of such graphs are either disconnected or contain a clique of size n/2 ). In this paper we discuss the minimizing graphs of a special class of graphs of order n whose complements are connected and contains exactly one cycle (namely the class Ucn of graphs whose complements are unicyclic), and characterize the unique minimizing graph in Ucn when n ≥ 20. [ABSTRACT FROM AUTHOR]

Letbe a signed graph and letbe the adjacency matrix of. The nullity ofis the multiplicity of eigenvalue zero in the spectrum of. In this paper, we characterize the signed graphs of orderwith nullityor, and introduce a graph transformation which preserves the nullity. As an application we determine the unbalanced bicyclic signed graphs of orderwith nullityor, and signed bicyclic graphs (including simple bicyclic graphs) of orderwith nullity. [ABSTRACT FROM AUTHOR]

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]

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 edges. [ABSTRACT FROM AUTHOR]

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: 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 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: 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: 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: 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]