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Abstract: This paper studies the problem of estimating the spectral radius of trees with the given number of vertices and maximum degree. We obtain the new upper bounds on the spectral radius of the trees, and the results are the best upper bounds expressed by the number of vertices and maximum degree, at present. Let be a tree on n vertices with maximum degree Δ, where . Denote by the spectral radius of T. We prove that [(1)] if , then , and equality holds if and only if T is an almost completely full-degree tree of 3 levels; [(2)] if , then , and equality holds if and only if T is a completely full-degree tree of 3 levels; [(3)] if , then , where . [Copyright &y& Elsevier]

Abstract: The rank of a graph is that of its adjacency matrix. A graph is called reduced if it has no isolated vertices and no two vertices with the same set of neighbors. We determine the maximum order of reduced trees as well as bipartite graphs with a given rank and characterize those graphs achieving the maximum order. [Copyright &y& Elsevier]

Abstract: We determine all trees whose second largest eigenvalue does not exceed . Next, we consider two classes of bipartite graphs, regular and semiregular, with small number of distinct eigenvalues. For all graphs considered we determine those whose second largest eigenvalue is equal to . Some additional results are also given. [Copyright &y& Elsevier]

Abstract: We characterize the eigenvalues and energy of the line graph whenever is (i) a generalized Bethe tree, (ii) a Bethe tree, (iii) a tree defined by generalized Bethe trees attached to a path, (iv) a tree defined by generalized Bethe trees having a common root, (v) a graph defined by copies of a generalized Bethe tree attached to a cycle and (vi) a graph defined by copies of a star attached to a cycle; in this case, explicit formulas for the eigenvalues and energy of are derived. [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: A Bethe tree of levels, , is a rooted tree such that the root vertex has degree , the vertices from level to have degree and the vertices at level are leaves. In this paper, we obtain a recurrence relation for the characteristic polynomial and the Laplacian characteristic polynomial of Bethe trees. As an application, we prove that there are no integral Bethe trees except for the star , and we obtain a recurrence relation for the Laplacian-energy-like invariant of Bethe trees. [Copyright &y& Elsevier]

Abstract: A semiregular tree is a tree where all non-pendant vertices have the same degree. Among all semiregular trees with fixed order and degree, a graph with minimal (adjacency/Laplacian) spectral radius is a caterpillar. Counter examples show that the result cannot be generalized to the class of trees with a given (non-constant) degree sequence. [Copyright &y& Elsevier]