On the Aσ-spectral radii of graphs with some given parameters

Abstract

Given a graph $G$, the adjacency matrix and degree diagonal matrix of $G$ are denoted by $A(G)$ and $D(G)$, respectively. In 2017, Nikiforov proposed the $A_{\sigma }$-matrix: $A_{\sigma }(G)=\sigma D(G)+(1-\sigma )A(G)$, where $\sigma \in [0,1]$. The largest eigenvalue of this novel matrix is called the $A_{\sigma }$-index of $G$. Let ${\mathcal{ℬ}}_n^{\alpha }$ be the class of $n$-vertex block graphs with independence number $\alpha$ and let $\mathcal{𝒢}(n,k)$ be another class of $n$-vertex graphs with $k$ cut edges. We show that the maximum $A_{\sigma }$-index, among all graphs $G\in {\mathcal{ℬ}}_n^{\alpha }$ (resp. $G\in \mathcal{𝒢}(n,k)$), is attained at a unique graph. It is surprising to see that in both cases, the extremal graphs are usually pineapple graphs. We use two methods to establish upper bounds on the $A_{\sigma }$-index of the corresponding extremal graphs. As a byproduct we obtain an upper bound for signless Laplacian spectral radius $q_1(G)$, when $G\in {\mathcal{ℬ}}_n^{\alpha }$.