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Let G be a graph with adjacency matrix A (G) and D (G) be the diagonal matrix of the degrees of G. Nikiforov (2017) defined the matrix A α (G) as A α (G) = α D (G) + (1 − α) A (G) , α ∈ [ 0 , 1 ]. The A α -spectral radius is the largest eigenvalue of A α (G). Let G m d be the set of all connected graphs with size m and diameter d. In this paper, we determine the graphs with maximal A α -spectral radius among all graphs in G m d. [ABSTRACT FROM AUTHOR]

An edge labeling of a graph G = (V , E) is said to be local antimagic if it is a bijection f : E → { 1 , ... , | E | } such that for any pair of adjacent vertices x and y , f + (x) ≠ f + (y) , where the induced vertex label of x is f + (x) = ∑ e ∈ E (x) f (e) (E (x) is the set of edges incident to x). The local antimagic chromatic number of G , denoted by χ la (G) , is the minimum number of distinct induced vertex labels over all local antimagic labelings of G. In this paper, several sufficient conditions to determine the local antimagic chromatic number of the join of graphs are obtained. We then determine the exact value of the local antimagic chromatic number of many join graphs. [ABSTRACT FROM AUTHOR]

Let G n , k D be the set of all k -connected graphs of order n with diameter D. In this paper, we determine the graphs with maximal signless Laplacian spectral radius among all graphs in G n , k D. [ABSTRACT FROM AUTHOR]

An edge labeling of a connected graph G = (V , E) is said to be local antimagic if it is a bijection f : E → { 1 , ... , | E | } such that for any pair of adjacent vertices x and y, f + (x) ≠ f + (y) , where the induced vertex label f + (x) = ∑ f (e) , with e ranging over all the edges incident to x. The local antimagic chromatic number of G, denoted by χ la (G) , is the minimum number of distinct induced vertex labels over all local antimagic labelings of G. In this paper, we give counterexamples to the lower bound of χ la (G ∨ O 2) that was obtained in [Local antimagic vertex coloring of a graph, Graphs Combin. 33:275–285 (2017)]. A sharp lower bound of χ la (G ∨ O n) and sufficient conditions for the given lower bound to be attained are obtained. Moreover, we settled Theorem 2.15 and solved Problem 3.3 in the affirmative. We also completely determined the local antimagic chromatic number of complete bipartite graphs. [ABSTRACT FROM AUTHOR]

Abstract: Let be the second smallest normalized Laplacian eigenvalue of a graph . In this paper, we investigate the behavior on when the graph is perturbed by separating an edge. This result can be used to determine all trees and unicyclic graphs with . Moreover, the trees and unicyclic graphs with are also determined, respectively. [Copyright &y& Elsevier]

Abstract: The normalized algebraic connectivity of a graph G, denoted by , is the second smallest eigenvalue of its normalized Laplacian matrix. In this paper, we firstly determine all trees with . Then we classify such trees into six classes and prove that for , where and . At the same time, the values of the normalized algebraic connectivity for the six classes of trees are provided, respectively. These results are similar to those on the algebraic connectivity which were obtained by Yuan et al. (2008) [8]. [Copyright &y& Elsevier]

Abstract: Let G be a simple graph with n vertices, m edges. If each edge of G belongs to t triangles ( ), then we present a new upper bound for the Laplacian spectral radius of G which improves some known upper bounds. [Copyright &y& Elsevier]

Abstract: The second largest Laplacian eigenvalue of a graph is the second largest eigenvalue of the associated Laplacian matrix. In this paper, we study extremal graphs for the extremal values of the second largest Laplacian eigenvalue and the Laplacian separator of a connected graph, respectively. All simple connected graphs with second largest Laplacian eigenvalue at most 3 are characterized. It is also shown that graphs with second largest Laplacian eigenvalue at most 3 are determined by their Laplacian spectrum. Moreover, the graphs with maximum and the second maximum Laplacian separators among all connected graphs are determined. [Copyright &y& Elsevier]

Abstract: An eigenvalue of a graph is called main if there is an associated eigenvector not orthogonal to , the vector with each entry equal to 1. In this work, an error in a prior paper [Y. Hou and F. Tian, Unicyclic graphs with exactly two main eigenvalues, Appl. Math. Letters, 19 (2006), 1143–1147] is pointed out and the properties of the graphs with exactly two main eigenvalues and with pendent vertices are discussed. As an application, we obtain, together with known results, all connected bicyclic and tricyclic graphs with exactly two main eigenvalues. [Copyright &y& Elsevier]

Abstract: The Laplacian spectral radius of a graph G is the largest eigenvalue of its Laplacian matrix. In this paper, the first three smallest values of the Laplacian spectral radii among all connected graphs with clique number are obtained. [Copyright &y& Elsevier]

By the well-known Perron–Frobenius Theorem [3] , for a connected graph G , its largest eigenvalue strictly increases when an edge is added. We are interested in how the other eigenvalues of a connected graph change when edges are added. Examples show that all cases are possible: increased, decreased, unchanged. In this paper, we consider the effect on the eigenvalues by suitably adding edges in particular families, say the family of connected graphs with clusters. By using the result, we also consider the effect on the energy by suitably adding edges to the graphs of the above families. [ABSTRACT FROM AUTHOR]