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On the revised edge-Szeged index of graphs. (English) Zbl 1464.92311

Summary: The revised edge-Szeged index of a connected graph \(G\) is defined as \(\mathrm{Sz}_e^*(G)=\sum_{e=uv \in E(G)}\left(m_u(e|G)+\frac{(m_0(e|G)}{2}\right)\left(m_v(e|G)+\frac{(m_0(e|G)}{2}\right)\), where \(m_u(e|G)\), \(m_v(e|G)\) and \(m_0(e|G)\) are, respectively, the number of edges of \(G\) lying closer to vertex \(u\) than to vertex \(v\), the number of edges of \(G\) lying closer to vertex \(v\) than to vertex \(u\), and the number of edges equidistant to \(u\) and \(v\). In this paper, we give an effective method for computing the revised edge-Szeged index of unicyclic graphs and using this result we identify the minimum revised edge-Szeged index of conjugated unicyclic graphs (i.e., unicyclic graphs with a perfect matching). We also give a method of calculating revised edge-Szeged index of the joint graph.

MSC:

92E10 Molecular structure (graph-theoretic methods, methods of differential topology, etc.)
05C92 Chemical graph theory
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