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The Zagreb eccentric vertex degree indices of nanotubes and nanotori. (English) Zbl 1358.05067
Summary: The eccentric vertex degree of a vertex \(v\) of a simple connectedgraph \(G\), \(e_v\), is defined as: \(e_v=\max\{d_{u_1},d_{u_2},\ldots,d_{u_n}\}\), where \(d_{u_i}\)enotes the degree of the vertex of \(u_i\) which is one of the furthest vertices from \(v\). The total eccentric vertex degree is defined as: \(TE=\sum_{v\in E(G)}e_v\), where \(e_v\) denotes the eccentric degree of the vertex \(v\). The first Zagreb eccentric vertex degree alpha index is defined as: \(DM^\alpha_1=\sum_{v\in E(G)}e_v^2\). The first Zagreb eccentric vertex degree beta index is defined as: \(DM_1^\beta=\sum_{uv\in E(G)}(e_u+e_v)\). And the second Zagreb eccentric vertex degree index is defined as: \(DM_2=\sum_{uv\in E(G)}e_ue_v\). In this study, we compute the exact value of the Zagreb eccentric vertex degree indices of \(TUC_4C_8(S)\) nanotubes and \(TC_4C_8(S)\) nanotori.
MSC:
05C07 Vertex degrees
05C12 Distance in graphs
05C90 Applications of graph theory
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