The Eccentric Distance Sum, the Harary Index, and the Degree Powers of Graphs with Given Diameter

Abstract

Let \(G\) be a simple connected graph with the vertex set \(V(G)\). The eccentric distance sum of \(G\) is defined as \(\xi^d(G) = \sum_{v \in V(G)} \varepsilon(v) D_G(v)\), where \(\varepsilon(v)\) is the eccentricity of the vertex \(v\) and \(D_G(v)\) is the sum of all distances from the vertex \(v\). The Harary index of \(G\) is defined as \(H(G) = \sum_{u,v \in V(G)} \frac{1}{d(u, v)}\), where \(d(u, v)\) is the distance between \(u\) and \(v\) in \(G\). The degree powers of \(G\) is defined as \(H(G) = \sum_{|u,v| \subseteq V(G)} \frac{1}{d(u,v)}\) for the natural number \(p \geq 1\). In this paper, we determine the extremal graphs with the minimal eccentric distance sum, the maximal Harary index, and the maximal degree powers among all graphs with given diameter.