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On certain topological indices of nanostructures using \(Q(G)\) and \(R(G)\) operators. (English) Zbl 1417.05035

Topological indices are numerical parameters of graphs, most commonly used in chemistry for predicting properties of some chemical compounds. In the present paper, the authors consider five topological indices of some graph operators. In particular, for a connected graph \(G\), the first Zagreb index of \(G\), \(M_1(G)\), and the second Zagreb index of \(G\), \(M_2(G)\), are defined as \(M_1(G) = \sum_{e=uv \in E(G)} \big(d_G(u) + d_G(v) \big)\) and \(M_2(G) = \sum_{e=uv \in E(G)} d_G(u)d_G(v)\), where \(d_G(u)\) and \(d_G(v)\) denote the degrees of vertices \(u\) and \(v\), respectively. On the other hand, the authors also consider the third Zagreb index, defined as \(M_3(G) = \sum_{e=uv \in E(G)} \big| d_G(u) - d_G(v) \big|.\)
However, in the literature this index is also known as the Albertson index (denoted by \(\mathrm{Alb}(G)\)) and it is used as an irregularity measure. Moreover, the hyper-Zagreb index of \(G\), \(HM(G)\), and the forgotten index of \(G\), \(F(G)\), are defined as \(HM(G) = \sum_{e=uv \in E(G)} \big(d_G(u) + d_G(v) \big)^2\) and \(F(G) = \sum_{e=uv \in E(G)} \big( d_G(u)^2 + d_G(v)^2 \big)\).
To state the main results, we need to define two graph operators, \(Q(G)\) and \(R(G)\), for a graph \(G\). The graph \(Q(G)\) is obtained from \(G\) in the following way: firstly we insert a new vertex into each edge of \(G\) and then we join any two new vertices that lie on adjacent edges. In addition, the graph \(R(G)\) is obtained from \(G\) by adding a new vertex corresponding to every edge of \(G\) and by joining each new vertex to the end vertices of the edge corresponding to it.
Finally, \(TUC_4C_8[p,q]\) denotes the \(C_4C_8\) lattice with exactly \(p\) octagons in each row and \(q\) octagons in each column. Furthermore, the corresponding nanotube and nanotorus are obtained by wrapping the lattice \(TUC_4C_8[p,q]\) and joining some vertices with new edges.
As the main result of this paper, the authors calculate closed formulas for the indices \(M_1(Q(G))\), \(M_2(Q(G))\), \(M_3(Q(G))\), \(HM(Q(G))\), \(F(Q(G))\), \(M_1(R(G))\), \(M_2(R(G))\), \(M_3(R(G))\), \(HM(R(G))\), and \(F(R(G))\), where \(G\) is the \(TUC_4C_8[p,q]\) lattice, the \(TUC_4C_8[p,q]\) nanotube, and the \(TUC_4C_8[p,q]\) nanotorus.

MSC:

05C07 Vertex degrees
05C40 Connectivity
05C90 Applications of graph theory
92E10 Molecular structure (graph-theoretic methods, methods of differential topology, etc.)
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