On the A_α-index of graphs with given order and dissociation number

Given a graph $G,$ a subset of vertices is called a maximum dissociation set of G if it induces a subgraph with vertex degree at most 1, and the subset has maximum cardinality. The cardinality of a maximum dissociation set is called the dissociation number of G. The adjacency matrix and the degree diagonal matrix of G are denoted by $A(G)$ and $D(G),$ respectively. In 2017, Nikiforov proposed the A_α-matrix: A_α(G)=α D(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 firstly determine the connected graph (resp. bipartite graph, tree) having the largest A_α-index over all connected graphs (resp. bipartite graphs, trees) with fixed order and dissociation number. Secondly, we describe the structure of all the n-vertex graphs having the minimum A_α-index with dissociation number τ, where τ2/3n. Finally, we identify all the connected n-vertex graphs with dissociation number τ∈\2,2/3n,n-1,n-2\ having the minimum A_α-index.