Minimizing vertex-degree function index for k-generalized quasi-unicyclic graphs

Keywords: Vertex-degree function index, general first Zagreb index, second multiplicative Zagreb index, sum lordeg index, k-generalized quasi-unicyclic graph, Jensen inequality


In this paper the problem of minimizing vertex-degree function index Hf(G) for k-generalized quasi-unicyclic graphs of order n is solved for k ≥ 1 and n ≥ 2k + 2 if the function f is strictly increasing and strictly convex. These conditions are fulfilled by general first Zagreb index 0Rα(G) if α > 1, second multiplicative Zagreb index ∏2(G) and sum lordeg index SL(G). The extremal graph is unique for k = 1, n = 4 and for k ≥ 2 and it consists from a path x1, x2, …, xn − 1 and a new vertex xn adjacent with xk, xk + 1 and xk + 2.