Bounds on the $A_{\alpha}$-spread of a graph

Abstract

Let $G$ be a simple undirected graph. For any real number $\alpha \in[0,1]$, Nikiforov defined the $A_{\alpha}$-matrix of $G$ as $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$, where $A(G)$ and $D(G)$ are the adjacency matrix and the degree diagonal matrix of $G$, respectively. The $A_{\alpha}$-spread of a graph is defined as the difference between the largest eigenvalue and the smallest eigenvalue of the associated $A_{\alpha}$-matrix. In this paper, some lower and upper bounds on $A_{\alpha}$-spread are obtained, which extend the results of $A$-spread and $Q$-spread. Moreover, the trees with the minimum and the maximum $A_{\alpha}$-spread are determined, respectively.

Sharp bounds on the $A_α$-index of graphs in terms of the independence number

“…Lin, Huang and Xue [23] characterized the extremal graph with maximum A α -index with fixed order and number of cut vertices, and the extremal tree which attains the maximum A α -index with fixed order and matching number. For more advances on the A αindex and some attractive feature of A α -matrices, one may be referred to [6,17,18,19,20,21,22,27,28,29,30,35,38] and the references cited in.…”

On A_\alpha–Eigenvalues of Graphs and Topological Indices

“…Detail about the A α -spectral radius can be found in [3,4,6,9,10,13,18,19,21,23,26,28,29,31,32,34]. For detail about the A α -spread, one may refer to [14,15]. Detail concerning the topological indices considered in this paper can be found in [2,11,20,33] and in the references therein.…”