ON THE FIRST AND SECOND ZAGREB INDICES OF QUASI UNICYCLIC GRAPHS

Abstract

Let G be a simple graph. The graph G is called a quasi unicyclic graph if there exists a vertex x ∈ V (G) such that G−x is a connected graph with a unique cycle. Moreover, the first and the second Zagreb indices of G denoted by M1(G) and M2(G), are the sum of deg (u) overall vertices u in G and the sum of deg(u) deg(v) of all edges uv of G, respectively. The first and the second Zagreb indices are defined relative to the degree of vertices. In this paper, sharp upper and lower bounds for the first and the second Zagreb indices of quasi unicyclic graphs are given. 1. Basic Definitions The first and the second Zagreb indices are among the oldest topological indices defined in 1972 by Gutman and Trinajstić [9]. These numbers have been used to study the molecular complexity, chirality and some other chemical quantities. The first Zagreb index is defined as the sum of the squares of the degrees of the vertices, i.e. M1 (G) = ∑ u∈V (G) deg (u) and the second Zagreb index is the sum of deg(u)deg(v) overall edges uv of G. This means that M2 (G) = ∑ uv∈E(G) deg(u)deg(v). The first and the second Zagreb indices are defined relative to the degree of vertices, which we summarize them without referring to the degree of vertices. MSC(2010): Primary: 05C35; Secondary: 05C07.