Document Type : Original Article

**Author**

Department of Mathematics, Iran University of Science and Technology (IUST), Narmak, Tehran 16844, Iran

**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 $M_1(G)=\sum_{uv\in E}d_u+d_v$ and $M_1(G)=\sum_{uv\in E}d_ud_v$, where du denotes the degree of vertex u in G. Recently, we know new versions of Zagreb indices as $M_1^{*}(G)=\sum_{uv\in E}ecc(u)+ecc(v)$, $M_1^{**}(G)=\sum_{u\in V}ecc(u)^2$ and $M_2^{*}(G)=\sum_{uv\in E}ecc(u)ecc(v)$, 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.

**Graphical Abstract**

**Keywords**

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June 2012

Pages 15-20

**Receive Date:**05 October 2011**Revise Date:**10 January 2012**Accept Date:**05 January 2012**First Publish Date:**01 June 2012