The lower bound of revised edge-Szeged index of unicyclic graphs with given diameter

Abstract

Given a connected graph G, the revised edge-revised Szeged index is defined as S{z}_e^{\ast }\left(G\right)=\sum _{e=uv\in E_G}\left(m_u\right(e)+\frac{m_0\left(e\right)}{2})\left(m_v\right(e)+\frac{m_0\left(e\right)}{2}), where m_u\left(e\right), m_v\left(e\right) and m_0\left(e\right) are 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 of G at the same distance to u and v, respectively. In this paper, by transformation and calculation, the lower bound of revised edge-Szeged index of unicyclic graphs with given diameter is obtained, and the extremal graph is depicted.

Key words： Wiener index; revised edge Szeged index; unicyclic graph; extremal graph

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