A new version of Zagreb index of circumcoronene series of benzenoid

Abstract

Among topological indices, Zagreb indices are very important, very old and they have many useful properties in chemistry and specially in mathematics chemistry. First and second Zagreb indices have been introduced by Gutman and Trinajstić as $M_1(G)=sum_{uvin E}d_u+d_v$ and $M_1(G)=sum_{uvin E}d_ud_v$, where du denotes the degree of vertex u in G. Recently, we know new versions of Zagreb indices as $M_1^{*}(G)=sum_{uvin E}ecc(u)+ecc(v)$, $M_1^{**}(G)=sum_{uin V}ecc(u)^2$ and $M_2^{*}(G)=sum_{uvin E}ecc(u)ecc(v)$, where ecc(u) is the largest distance between u and any other vertex v of G. In this paper, we focus one of these new topological indices that we call fifth Zagreb index $M_2^*(G)=M_5(G)$ and we compute this index for a famous molecular graph Circumcoronene series of benzenoid H_k, k≥ 1.