Extremal bipartite graphs with given parameters on the resistance-Harary index (Q2337997)

Summary: Resistance distance is a concept developed from electronic networks. The calculation of resistance distance in various circuits has attracted the attention of many engineers. This report considers the resistance-based graph invariant, the Resistance-Harary index, which represents the sum of the reciprocal resistances of any vertex pair in the figure \(G\), denoted by \(R H(G)\). Vertex bipartiteness in a graph \(G\) is the minimum number of vertices removed that makes the graph \(G\) become a bipartite graph. In this study, we give the upper bound and lower bound of the \(R H\) index, and describe the corresponding extremal graphs in the bipartite graph of a given order. We also describe the graphs with maximum \(R H\) index in terms of graph parameters such as vertex bipartiteness, cut edges, and matching numbers.