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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: Haemers et al. [W.H. Haemers, X.G. Liu, Y.P. Zhang, Spectral characterizations of lollipop graphs, Linear Algebra Appl. 428 (2008) 2415–2423] first investigated the spectral characterizations of the so called lollipop graphs with order , which is obtained by identifying a vertex of a cycle with order and a pendent vertex of a path with order . For the graphs with least eigenvalue at least , Cvetković and Lepović [D. Cvetković, M. Lepović, Cospectral graphs with least eigenvalue at least , Publ. Inst. Math., Nouv. Sér. 78(92) (2005) 51–63] introduced the discriminant and the star value of a graph, whose relations are investigated in this paper. Employing this relation and other techniques, we prove that all line graphs of lollipop graphs are determined by their adjacency spectra. [Copyright &y& Elsevier]

Abstract: The rank of a graph G is defined to be the rank of its adjacency matrix. In this paper, we consider the following problem: What is the structure of a connected graph with rank 4? This question has not yet been fully answered in the literature, and only some partial results are known. In this paper we resolve this question by completely characterizing graphs G whose adjacency matrix has rank 4. [Copyright &y& Elsevier]

Abstract: The spectral excess theorem provides a quasi-spectral characterization for a (regular) graph with distinct eigenvalues to be distance-regular graph, in terms of the excess (number of vertices at distance ) of each of its vertices. The original approach, due to Fiol and Garriga in 1997, was obtained by using a local approach, so giving a characterization of the so-called pseudo-distance-regularity around a vertex. In this paper we present a new simple projection method based in a global point of view, and where the mean excess plays an essential role. [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]

This part of our work further extends our project of building a new spectral theory of graphs (based on the signless Laplacian) by some results on graph angles, by several comments and by a short survey of recent results. [ABSTRACT FROM AUTHOR]

Abstract: In J.F. Wang and S.N. Shu (2012) [2], the authors lost the condition that the line graph does not contain the cycle of order 4 in Lemma 3.4, which makes their main result (i.e., Theorem 3.1) of that paper not completely correct. Here, some comments are given. [Copyright &y& Elsevier]