A note on some bounds of the \(\alpha\)-Estrada index of graphs (Q2246532)

Summary: Let \(G\) be a simple graph with \(n\) vertices. Let \(\widetilde{A}_\alpha(G)=\alpha D(G)+(1 - \alpha)A(G)\), where \(0\leq\alpha\leq1\) and \(A(G)\) and \(D(G)\) denote the adjacency matrix and degree matrix of \(G\), respectively. \(\mathrm{EE}_\alpha(G)= \sum_{i = 1}^n e^{\lambda_i}\) is called the \(\alpha\)-Estrada index of \(G\), where \(\lambda_1,\dots, \lambda_n\) denote the eigenvalues of \(\widetilde{A}_\alpha(G)\). In this paper, the upper and lower bounds for \(\mathrm{EE}_\alpha(G)\) are given. Moreover, some relations between the \(\alpha\)-Estrada index and \(\alpha\)-energy are established.