Research ArticleNo Access

General eccentric distance sum of graphs with given diameter

Tomáš Vetrík

Department of Mathematics and Applied Mathematics, University of the Free State, Bloemfontein, South Africa

https://doi.org/10.1142/S1793557123500572Cited by:0 (Source: Crossref)

Abstract

For $a, b \in ℝ$ , the general eccentric distance sum of a connected graph $G$ is defined as ${EDS}_{a, b} (G) = \sum_{u \in V (G)} {[{ecc}_{G} (u)]}^{a} {[D_{G} (u)]}^{b}$ , where $V (G)$ is the vertex set of $G$ , ${ecc}_{G} (u)$ is the eccentricity of $u \in V (G)$ , $D_{G} (u) = \sum_{v \in V (G)} d_{G} (u, v)$ and $d_{G} (u, v)$ is the distance between vertices $u$ and $v$ in $G$ . For $a \geq 0$ and $b \geq 1$ , we present the graphs having the smallest general eccentric distance sum among graphs with given order and diameter, and among bipartite graphs with given order and odd diameter. The extremal graphs for the classical eccentric distance sum are corollaries of our results on the general eccentric distance sum.

Communicated by I. Peterin

Keywords:

AMSC: 05C09, 05C12, 92E10

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