Some observations on comparing Zagreb indices

Abstract

Let G be a simple graph possessing n vertices and m edges. Let di be the degree of the i-th vertex of G , i = 1, . . . , n . The first Zagreb index M1 is the sum of di2 over all vertices of G . The second Zagreb index M2 is the sum di dj over pairs of adjacent vertices of G . In this paper we search for graph for which M1 /n = M2 /m , and show how numerous such graphs can be constructed. In addition, we find examples of graphs for which M1 /n > M2 /m , which are counterexamples for the earlier conjectured inequality M1 /n ≤ M2 /m .