On the A_-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_-matrix: A_ (G) = D (G) + (1-) A (G), where 0, 1. The largest eigenvalue of this novel matrix is called the A_-index of G. In this paper, we firstly determine the connected graph (resp. bipartite graph, tree) having the largest A_-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_-index with dissociation number, where 23n. Finally, we identify all the connected n-vertex graphs with dissociation number \2, 2{3n, n-1, n-2\} having the minimum A_-index.