Connected graphs of fixed order and size with maximal Aα-index: The one-dominating-vertex case

Abstract

For any real number α∈[0,1], by the Aα-matrix of a graph G we mean the matrix Aα(G)=αD(G)+(1−α)A(G), where A(G) and D(G) denote respectively the adjacency matrix and the diagonal matrix of vertex degrees of G. The largest eigenvalue of Aα(G) is called the Aα-index of G. Chang and Tam (2011) have proved that for every pair of integers n,k with −1≤k≤n−3, Hn,k, the graph obtained from the star K1,n−1 by joining a vertex of degree 1 to k+1 other vertices of degree 1, is the unique connected graph that maximizes the Q-index (i.e., the signless Laplacian spectral radius or, equivalently, the A12-index) over all connected graphs with n vertices and n+k edges. In this paper it is proved that for every pair of integers n,k with −1≤k≤n−3, when 12<α<1 or α=12 and k≠2, the graph Hn,k is the unique connected graph that maximizes the Aα-index over all connected graphs with n vertices and n+k edges. This work extends (and also provides an alternative proof for) the above-mentioned result of Chang and Tam. A complete overview of the history of the maximal index problems is also given.