On harmonic centers of graphs

Abstract: The harmonic centrality of a vertex $v$ in a graph $G = (V,E)$ is the sum of reciprocals of distances of vertices of $G$ from $v$. The vertices of $G$ which have the maximum (minimum) harmonic centrality form the harmonic center (or periphery, resp.) of $G$. We study harmonic centers of graphs and their localization in graph blocks, presenting sufficient conditions for graphs (in terms of diameter or number of edges) to have those centers contained in a single block; in addition, we show that each connected graph is the harmonic center as well as harmonic periphery of some graphs.