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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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