Extremal vertex-degree function index for trees and unicyclic graphs with given independence number

Abstract

In this paper the problem of maximizing vertex-degree function index Hf(G) for trees and unicyclic graphs G of order n and independence number s is solved for strictly convex functions f(x). In the case of unicyclic graphs f(x) must also satisfy strict inequality f(4)+3f(2)>3f(3)+f(1). These conditions are fulfilled by general first Zagreb index 0Rα(G) for α>2, second multiplicative Zagreb index ∏2(G) and sum lordeg index SL(G). The extremal graphs are unique and they are stars or have diameter equal to three or to four. The same results are valid for the corresponding minimization problem when f(x) is strictly concave and the inequality is reversed.