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Total eccentricity index of trees with fixed pendent vertices and trees with fixed diameter. (English) Zbl 1520.05026

Summary: Let \(\mathcal{G}=(\mathcal{V}_{\mathcal{G}},\mathcal{E}_{\mathcal{G}})\) be a connected graph with \(n\) vertices. The eccentricity \(\operatorname{ec}_{\mathcal{G}}(w)\) of a vertex \(w\) in \(\mathcal{G}\) is the maximum distance between \(w\) and any other vertex of \(\mathcal{G} \). The total eccentricity index \(\tau (\mathcal{G})\) of \(\mathcal{G}\) is defined as \(\tau (\mathcal{G})=\sum \nolimits_{w\in \mathcal{V}_{\mathcal{G}}} \operatorname{ec}_{\mathcal{G}}(w)\). In this paper, we derive the trees with minimum and maximum total eccentricity index among the class of \(n\)-vertex trees with \(p\) pendent vertices. We also determine the trees with minimum and maximum total eccentricity index among the class of \(n\)-vertex trees with a given diameter.

MSC:

05C09 Graphical indices (Wiener index, Zagreb index, Randić index, etc.)
05C12 Distance in graphs
05C05 Trees
05C35 Extremal problems in graph theory
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References:

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