On a Lower Bound for the Eccentric Connectivity Index of Graphs
| Autor: | Devsi Bantva |
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| Rok vydání: | 2018 |
| Předmět: | |
| Zdroj: | Algorithms and Discrete Applied Mathematics ISBN: 9783319741796 CALDAM |
| DOI: | 10.1007/978-3-319-74180-2_15 |
| Popis: | The eccentric connectivity index of a graph G, denoted by \(\xi ^{c}(G)\), defined as \(\xi ^{c}(G)\) = \(\sum _{v \in V(G)}\epsilon (v) \cdot \mathrm{d}(v)\), where \(\epsilon (v)\) and \(\mathrm{d}(v)\) denotes the eccentricity and degree of a vertex v in a graph G, respectively. The volcano graph \(V_{n,d}\) is a graph obtained from a path \(P_{d+1}\) and a set S of \(n-d-1\) vertices, by joining each vertex in S to a central vertex/vertices of \(P_{d+1}\). In [4], Morgan et al. proved that \(\xi ^{c}(G) \ge \xi ^{c}(V_{n,d})\) for any graph of order n and diameter \(d \ge 3\). In this paper, we present a short and simple proof of this result by considering the adjacency of vertices in graphs. |
| Databáze: | OpenAIRE |
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