On a relation between the Zagreb indices

Abstract

Let G=(V,E) be a simple graph with n=|V| vertices and m=|E| edges. The first and the second Zagreb index are defined as M1= σuεVd2u and M2= σuvεEdudv, where du is the degree of vertex u. Professor Pierre Hansen at the International Academy of Mathematical Chemistry Meeting in 2006 conjectured that M1/m≤ M1/n holds for all simple graphs. While the conjecture is true for trees, unicyclic and chemical graphs, several counterexamples appeared in the literature. Here we extend the construction of counterexamples by showing that we may add a sufficiently large star to any graph G with m≥n+δ to obtain a counterexample For the variable Zagreb indices λ M1= σuεVdu2λ and λ M2=σuvεvd 2λuand λM2= σuvεEdλudλv, we prove that any graph G can be extended by a suitably large star so that λM1/n>λM2/m when 0<λ<1, and λM1/n<lambda;M2/m when λ<0 or λ>1.