Short communication, Note

https://doi.org/10.5562/cca3028

Comparison Between Two Eccentricity-based Topological Indices of Graphs

Kexiang Xu ; College of Science, Nanjing University of Aeronautics & Astronautics, Nanjing, Jiangsu 210016, PR China
Xia Li ; College of Science, Nanjing University of Aeronautics & Astronautics, Nanjing, Jiangsu 210016, PR China

Abstract

For a connected graph \(G\), the eccentric connectivity index (ECI) and the first Zagreb eccentricity index of \(G\) are defined as \( \xi ^{c}(G)= \sum_{v_i \in V(G)}\deg_G(v_i)\varepsilon_G(v_i)\) and \(E_1(G)=\sum_{v_i\in V(G)}\varepsilon_{G}(v_i)^{2}\), respectively, where \(\deg_G(v_i)\) is the degree of \(v_i\) in \(G\) and \(\varepsilon_G(v_i)\) denotes the eccentricity of vertex \(v_i \)in \(G\). In this paper we compare the eccentric connectivity index and the first Zagreb eccentricity index of graphs. It is proved that \(E_1(T)>\xi^c(T)\) for any tree \(T\). This improves a result by Das[25] for the chemical trees. Moreover, we also show that there are infinite number of chemical graphs \(G\) with \(E_1(G)>\xi^c(G)\). We also present an example in which infinite graphs \(G\) are constructed with \(E_1(G)=\xi^c(G)\) and give some results on the graphs \(G\) with \(E_1(G)<\xi^c(G)\). Finally, an effective construction is proposed for generating infinite graphs with each comparative inequality possibility between these two topological indices.

Keywords

Graph; First Zagreb eccentricity index; Eccentric connectivity index

Hrčak ID:

181033

URI

https://hrcak.srce.hr/181033

Publication date:

19.12.2016.

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