Further properties of the second Zagreb index

Abstract

We determine the maximum value of the second Zagreb index among graphs with given numbers of vertices, edges, minimum and maximum vertex degrees. As the extremal graphs may happen to be multigraphs, we, in addition, determine trees with the maximum value of the second Zagreb index among all trees with given number of vertices and maximum vertex degree. As an application of the latter result, we answer the question of Ashrafi, Do¡slíc and Hamzeh [MATCH Commun. Math. Comput. Chem. 65 (2011), 85-92] on the minimum values of Zagreb coindices over chemical trees.