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The energy of a simple graph G , E ( G ) , is the sum of the absolute values of the eigenvalues of its adjacency matrix. The energy of line graph and the signless Laplacian energy of graph G are denoted by E ( L G ) ( L G is the line graph of G ) and LE + ( G ) , respectively. In this paper we obtain a relation between E ( L G ) and LE + ( G ) of graph G . From this relation we characterize all the graphs satisfying E ( L G ) = LE + ( G ) + 4 m n − 4 . We also present a relation between E ( G ) and E ( L G ) . Moreover, we give an upper bound on E ( L G ) of graph G and characterize the extremal graphs. [ABSTRACT FROM AUTHOR]

Abstract: The Randić index R of a graph G is defined as the sum of over all edges of G, where denotes the degree of a vertex in G. is the largest eigenvalue of the signless Laplacian matrix of G, where D is the diagonal matrix with degrees of the vertices on the main diagonal and A is the adjacency matrix of G. Hansen and Lucas [18] conjectured (1) and equality holds for and (2) with equality if and only if for and for , respectively. In this paper, we prove the conjecture (1) and obtain a result very close to the conjecture (2). Moreover, we give some results relating harmonic index and the largest eigenvalue of the adjacency matrix. [Copyright &y& Elsevier]

Abstract: Let be two simple connected graphs. Denote the corona and the edge corona of by and , respectively. In this paper, we first introduce a new invariant, the -coronal of a graph matrix , where the matrix is associated with a graph in a prescribed way. Then it is used to compute the signless Laplacian spectrum of and in terms of the signless Laplacian spectrum of and . In addition, the spectrum of are also given in terms of the spectrum of and . Finally, as an application of these results, we construct many pairs of nonisomorphic signless Laplacian cospectral graphs. [Copyright &y& Elsevier]

Abstract: A -graph is a graph consisting of two cycles and with just a vertex in common. Fan and Luo in Theorem 4.4 alleged that a -graph is determined by its adjacency spectrum if it contains no cycle . However, according to Theorem 6.3 we find that the result in Theorem 4.4 is not completely correct. In this paper, we prove that is DAS if and only if , and has a unique cospectral mate if . [Copyright &y& Elsevier]

Abstract: A generalized Bethe tree is a rooted tree in which vertices at the same level have the same degree. For let be a generalized Bethe tree of levels and let such that (1) the edges of connecting vertices at consecutive levels have the same weight, and (2) for , each set of children of at the level defines a clique in which the edges have weight For let be the graph obtained from and the cliques at the levels for all Let be the graph obtained from the graphs joined at their respective roots. We give a complete characterization of the eigenvalues, including their multiplicities, of the Laplacian, signless Laplacian and adjacency matrices of the graph . Finally, we characterize the normalized Laplacian eigenvalues when is an unweighted graph. [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]