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 Aσ-matrix: Aσ(G)=σD(G)+(1−σ)A(G), where σ∈[0,1]. The largest eigenvalue of this novel matrix is called the Aσ-index of G. Let ℬnα be the class of n-vertex block graphs with independence number α and let 𝒢(n,k) be another class of n-vertex graphs with k cut edges. We show that the maximum Aσ-index, among all graphs G∈ℬnα (resp. G∈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 Aσ-index of the corresponding extremal graphs. As a byproduct we obtain an upper bound for signless Laplacian spectral radius q1(G), when G∈ℬnα.