Sharp Bounds on the Aα-index of Graphs in Terms of the Independence Number

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^[24] proposed the A_α-matrix: A_α(G) = αD(G) + (1 − α)A(G), where α ∈ [0, 1]. The largest eigenvalue of this novel matrix is called the A_α-index of G. In this paper, we characterize the graphs with minimum A_α-index among n-vertex graphs with independence number i for α ∈ [0, 1), where \(i = 1,\,\,\left\lfloor {{n \over 2}} \right\rfloor,\left\lceil {{n \over 2}} \right\rceil,\,\left\lfloor {{n \over 2}} \right\rfloor + 1,n - 3,n - 2,n - 1\), whereas for i = 2 we consider the same problem for \(\alpha \in [0,{3 \over 4}]\). Furthermore, we determine the unique graph (resp. tree) on n vertices with given independence number having the maximum A_α-index with α ∈ [0, 1), whereas for the n-vertex bipartite graphs with given independence number, we characterize the unique graph having the maximum A_α-index with \(\alpha \in [{1 \over 2},1)\).