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Harmonic centrality calculates the importance of a node in a network by adding the inverse of the geodesic distances of this node to all the other nodes. Harmonic centralization, on the other hand, is the graph-level centrality score based on the node-level harmonic centrality. In this paper, we present some results on both the harmonic centrality and harmonic centralization of graphs resulting from some graph products such as Cartesian and direct products of the path $P_2$ with any of the path $P_m$, cycle $C_m$, and fan $F_m$ graphs., Comment: 10 pages, 3 figures

Centrality describes the importance of nodes in a graph and is modeled by various measures. Its global analogue, called centralization, is a general formula for calculating a graph-level centrality score based on the node-level centrality measure. The latter enables us to compare graphs based on the extent to which the connections of a given network are concentrated on a single vertex or group of vertices. One of the measures of centrality in social network analysis is harmonic centrality. It sums the inverse of the geodesic distances of each node to other nodes where it is 0 if there is no path from one node to another, with the sum normalized by dividing it by $m-1$, where $m$ is the number of nodes of the graph. In this paper, we present some results regarding the harmonic centralization of some important families of graphs with the hope that formulas generated herein will be of use when one determines the harmonic centralization of more complex graphs., Comment: 21 pages, 5 figures. arXiv admin note: text overlap with arXiv:2111.12239

One of the more recent measures of centrality in social network analysis is the normalized harmonic centrality. A variant of the closeness centrality, harmonic centrality sums the inverse of the geodesic distances of each node to other nodes where it is 0 if there is no path from one node to another. It is then normalized by dividing it by m-1, where m is the number of nodes of the graph. In this paper, we present notions regarding the harmonic centrality of some important classes of graphs., Comment: 13 pages, 5 figures