Equitable and Non-Equitable Zagreb Indices of Graphs

Abstract

The Zagreb indices have been introduced more than forty four years ago by Gutman and Trinajestic as the sum of the squares of the degrees of the vertices, and the sum of the products of the degrees of pairs of adjacent vertices, respectively, \cite{za1}. In this paper, we introduce the first and second equitable and non-equitable Zagreb indices as $M_1^e(G)=\sum_{u\in V(G)}\big[deg_e(u)\big]^2$, $M_2^e(G)=\sum_{uv\in E(G)}deg_e(u)deg_e(v)$, $M_1^{ne}(G)=\sum_{u\in V(G)}\big[deg_{ne}(u)\big]^2$ and $M_2^{ne}(G)=\sum_{uv\in E(G)}deg_{ne}(u)deg_{ne}(v)$, respectively, where $deg_e(u)$ and $deg_{ne}(u)$ denotes the equitable and non-equitable degrees of vertex $u$. Exact values for wheel, firecracker and firefly graph families are obtained, some properties of the equitable and non-equitable Zagreb indices are established.