New upper bounds on Zagreb indices

Abstract

The first Zagreb index M1(G) is equal to the sum of squares of the degrees of the vertices, and the second Zagreb index M2(G) is equal to the sum of the products of the degrees of pairs of adjacent vertices of the underlying molecular graph G. In this paper we obtain an upper bound on the first Zagreb index M1(G) of G in terms of the number of vertices (n), number of edges (m), maximum vertex degree (Δ1), second maximum vertex degree (Δ2) and minimum vertex degree (δ). Using this result we find an upper bound on M2(G). Moreover, we present upper bounds on \({M_1(G)+M_1(\overline{G})}\) and \({M_2(G)+M_2(\overline{G})}\) in terms of n, m, Δ1, Δ2, δ, where \({\overline{G}}\) denotes the complement of G.