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

Abstract

For a graph G=(VG,EG), a subset S⊆VG is called a maximum dissociation set if the induced subgraph G[S] does not contain P3 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 Hf(G)=∑v∈VGf(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 Hf(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 Hf(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).