Extremal vertex-degree function index with given order and dissociation number

For a graph G = (V-G, E-G), a subset S subset of VG is called a maximum dissociation set if the induced subgraph G[S] does not contain P-3 as its subgraph, and the subset has maximum cardinality. The dissociation number of G is the number of vertices in a maximum dissociation set of G. This paper mainly studies the problem of determining the maximum values of the vertex-degree function index H-f(G) = & sum;(v is an element of VG)f(d(v)) and characterizing the corresponding extremal graphs among all trees and unicyclic graphs with fixed order and dissociation number when f(x) is a strictly convex function. Firstly, we describe all the trees having the maximum vertex-degree function index H-f(T) among trees with given order and dissociation number when f(x) is a strictly convex function. Then we determine the graphs having the maximum vertex-degree function index H-f(T) among unicyclic graphs with given order and dissociation number when f(x) is a strictly convex function and satisfies f (5)+ 2f (3) + f (2) > 3f (4) + f (1).(c) 2023 Elsevier B.V. All rights reserved.