Sharp bounds for the first Zagreb index and first Zagreb coindex

Abstract

Let G be an undirected connected graph with n ; n 2, vertices and m edges with vertex degrees D d 1 d 2 d n D . Lower and upper bounds of a graph invariants M 1 D P n i D1 d 2 i , referred to as the first Zagreb index, and N M 1 D P i oej .d i C d j /, named the first Zagreb coindex, depending on parameters n, m, and are obtained. The obtained results represent improvement of the results reported in the literature.