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Let G=(V,E) be a simple connected weighted graph on n vertices, in which the edge weights are positive definite matrices. The eigenvalues of G are the eigenvalues of its adjacency matrix. In this note, we present a correction in equality part in Theorem 2 [S. Sorgun, S. Büyükköse, The new upper bounds on the spectral radius of weighted graphs, Appl. Math. Comput. 218 (2012) 5231–5238]. In addition, some related results are also provided. [ABSTRACT FROM AUTHOR]

Abstract: Let denote the distance matrix of a connected graph with order , where is equal to the distance between and in . The largest eigenvalue of is called the distance spectral radius of graph , denoted by . In this paper, some graft transformations that decrease or increase are given. With them, for the graphs with both order and pendant vertices, the extremal graphs with the minimum distance spectral radius are completely characterized; the extremal graph with the maximum distance spectral radius is shown to be a dumbbell graph (obtained by attaching some pendant edges to each pendant vertex of a path respectively) when ; for , the extremal graphs with the maximum distance spectral radius are completely characterized. [Copyright &y& Elsevier]

Abstract: Let G be a simple graph with vertices , of degrees , respectively. Let A be the -adjacency matrix of G and D be the diagonal matrix diag. is called the signless Laplacian of G. The largest eigenvalue of is called the signless Laplacian spectral radius or Q-spectral radius of G. Denote by the chromatic number for a graph G. In this paper, for graphs with order n, the extremal graphs with both the given chromatic number and the maximal Q-spectral radius are characterized, the extremal graphs with both the given chromatic number and the minimal Q-spectral radius are characterized as well. [Copyright &y& Elsevier]

Abstract: A connected graph is called a quasi-k-cyclic graph, if there exists a vertex such that is a k-cyclic graph (connected with cyclomatic number k). In this paper we identify in the set of quasi-k-cyclic graphs (for ) those graphs whose spectral radius of the adjacency matrix (and the signless Laplacian if ) is the largest. In addition, for quasi-unicyclic graphs we identify as well those graphs whose spectral radius of the adjacency matrix is the second largest. [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]

Abstract: We consider the set of unicyclic graphs with prescribed degree sequence. In this set we determine the (unique) graph with the largest spectral radius (or index) with respect to the adjacency matrix. In addition, we give a conjecture about the (unique) graph with the largest index in the set of connected graphs with prescribed degree sequence. [Copyright &y& Elsevier]

Abstract: Let be a graph with vertices and be the largest eigenvalue of the adjacency matrix of . We study how large can be when does not contain cycles and paths of specified order. In particular, we determine the maximum spectral radius of graphs without paths of given length, and give tight bounds on the spectral radius of graphs without given even cycles. We also raise a number of open problems. [Copyright &y& Elsevier]