On the Average Eccentricity of Unicyclic Graphs

Abstract

Let \(G\) be a connected graph on \(n\) vertices. The average eccentricity of a graph \(G\) is defined as \(\varepsilon(G) = \frac{1}{n} \sum_{v \in V(G)} \varepsilon(v)\), where \(\varepsilon(v)\) is the eccentricity of the vertex \(v\), which is the maximum distance from it to any other vertex. In this paper, we characterize the extremal unicyclic graphs among \(n\)-vertex unicyclic graphs having the minimal and the second minimal average eccentricity.