Iranian Nanotechnology Society International Journal of Nanoscience and Nanotechnology 1735-7004 8 3 2012 09 01 The First and Second Zagreb Indices, First and Second Zagreb Polynomials of HAC5C6C7[p,q] and HAC5C7[p,q] Nanotubes 175 180 10754 EN M. R. Farahani Department of Applied Mathematics, Iran University of Science and Technology (IUST), Tehran, I. R. Iran Journal Article 2012 03 15 Topological indices are numerical parameters of a molecular graph G which characterize its topology. On the other hands, computing the connectivity indices of molecular graphs is an important branch in chemical graph theory. Therefore, we compute First Zagreb index Zg   <span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;"><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;">1</span></span><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: medium;"><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: medium;">(G)= </span></span><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;"><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;">uv E </span></span> <span style="font-family: TimesNewRomanPSMT; font-size: xx-small;"><span style="font-family: TimesNewRomanPSMT; font-size: xx-small;">(</span></span><em><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;"><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;">G </span></span></em><span style="font-family: TimesNewRomanPSMT; font-size: xx-small;"><span style="font-family: TimesNewRomanPSMT; font-size: xx-small;">) </span></span><span style="font-family: SymbolMT; font-size: large;" lang="JA"><span style="font-family: SymbolMT; font-size: large;" lang="JA">( </span></span><em><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;"><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;">u v </span></span></em><span style="font-family: SymbolMT; font-size: large;" lang="JA"><span style="font-family: SymbolMT; font-size: large;" lang="JA">)</span></span><span style="font-family: TimesNewRomanPSMT; font-size: medium;"><span style="font-family: TimesNewRomanPSMT; font-size: medium;">, </span></span><em><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: medium;"><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: medium;">d d </span></span></em><span style="font-family: SymbolMT; font-size: xx-small;" lang="JA"><span style="font-family: SymbolMT; font-size: xx-small;" lang="JA">∈ </span></span><span style="font-family: SymbolMT; font-size: x-large;" lang="JA"><span style="font-family: SymbolMT; font-size: x-large;" lang="JA">Σ </span></span><span style="font-family: SymbolMT; font-size: medium;" lang="JA"><span style="font-family: SymbolMT; font-size: medium;" lang="JA">+ </span></span><em><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: medium;"><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: medium;">Second Zagreb index</span></span></em> Zg   <span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;"><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;">2</span></span><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: medium;"><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: medium;">(G)= </span></span> <span style="font-family: SymbolMT; font-size: x-large;" lang="JA"><span style="font-family: SymbolMT; font-size: x-large;" lang="JA">( ) </span></span><em><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;"><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;">uv E </span></span></em><span style="font-family: TimesNewRomanPSMT; font-size: xx-small;"><span style="font-family: TimesNewRomanPSMT; font-size: xx-small;">(</span></span><em><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;"><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;">G </span></span></em><span style="font-family: TimesNewRomanPSMT; font-size: xx-small;"><span style="font-family: TimesNewRomanPSMT; font-size: xx-small;">) </span></span><em><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;"><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;">u v </span></span><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: medium;"><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: medium;">d d </span></span></em><span style="font-family: SymbolMT; font-size: xx-small;" lang="JA"><span style="font-family: SymbolMT; font-size: xx-small;" lang="JA">∈ </span></span><span style="font-family: SymbolMT; font-size: x-large;" lang="JA"><span style="font-family: SymbolMT; font-size: x-large;" lang="JA">Σ </span></span><span style="font-family: SymbolMT; font-size: medium;" lang="JA"><span style="font-family: SymbolMT; font-size: medium;" lang="JA">× </span></span><em><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: medium;"><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: medium;">and their polynomials Zg</span></span><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;"><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;">1</span></span><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: medium;"><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: medium;">(G,x)= </span></span></em><span style="font-family: TimesNewRomanPSMT; font-size: xx-small;"><span style="font-family: TimesNewRomanPSMT; font-size: xx-small;">( )</span></span> d   <span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;"><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;">u </span></span><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;"><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;">d</span></span><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;"><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;">v</span></span> uv E G   <span style="font-family: TimesNewRomanPS-ItalicMT;"><span style="font-size: medium;">x </span></span> <span style="font-family: SymbolMT; font-size: xx-small;" lang="JA"><span style="font-family: SymbolMT; font-size: xx-small;" lang="JA">∈</span></span> Σ   <span style="font-family: SymbolMT; font-size: xx-small;" lang="JA"><span style="font-family: SymbolMT; font-size: xx-small;" lang="JA">+ </span></span><em><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: medium;"><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: medium;">and Zg</span></span><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;"><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;">2</span></span><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: medium;"><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: medium;">(G,x)= </span></span></em><span style="font-family: TimesNewRomanPSMT; font-size: xx-small;"><span style="font-family: TimesNewRomanPSMT; font-size: xx-small;">( )</span></span> d   <span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;"><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;">u</span></span><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;"><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;">d</span></span><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;"><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;">v</span></span> uv E G   <span style="font-family: TimesNewRomanPS-ItalicMT;"><span style="font-size: medium;">x </span></span> <span style="font-family: SymbolMT; font-size: xx-small;" lang="JA"><span style="font-family: SymbolMT; font-size: xx-small;" lang="JA">∈ </span></span><span style="font-family: SymbolMT; font-size: x-large;" lang="JA"><span style="font-family: SymbolMT; font-size: x-large;" lang="JA">Σ </span></span><em><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: medium;"><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: medium;">for two</span></span></em> famous families of nanotubes “HAC   <span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;"><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;">5</span></span><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: medium;"><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: medium;">C</span></span><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;"><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;">6</span></span><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: medium;"><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: medium;">C</span></span><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;"><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;">7</span></span><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: medium;" lang="ZH-TW"><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: medium;" lang="ZH-TW">[p,q]” and “HAC</span></span><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;"><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;">5</span></span><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: medium;"><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: medium;">C</span></span><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;"><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: xx-small;">7</span></span><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: medium;" lang="ZH-TW"><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: medium;" lang="ZH-TW">[p,q]”.</span></span>     Molecular Graph Nanotubes First Zagreb index Second Zagreb index Zagreb polynomial