Extremal Bipartite Graphs with Given Parameters on the Resistance–Harary Index

Abstract

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.

Keywords

• Resistance–Harary index
• resistance distance
• cut edges
• bipartite graph
• matching number
• vertex bipartiteness