Multiplicative versions of first Zagreb index

Abstract

The first Zagreb index of a graph G, with vertex set V(G) and edge set E(G), is defined as M1(G) = Σuε∈V(G)d(u) 2 where d(u) denotes the degree of the vertex ε. An alternative expression for M1(G) is Σu∈εE(G)[d(u) + d(ε)]. We consider a multiplicative version of M1 defined as Π1(G) = Πuε∈E(G)[d(u) + d(v)]. We prove that among all connected graphs with a given number of vertices, the path has minimal Π1. We also determine the trees with the second-minimal Π1.