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.
Highlights
There are graph invariants that describe certain properties of a graph, which we call topological indices
We describe the graphs with maximum RH index in terms of graph parameters such as vertex bipartiteness, cut edges, and matching numbers
We describe the graphs with maximum RH index in terms of graph parameters such as vertex bipartiteness vb, cut edges, and matching numbers
Summary
There are graph invariants that describe certain properties of a graph, which we call topological indices. Finding the extreme value for the topological indices, as well as the related problem of characterizing the extremal graphs, attracted the attention of many researchers, and many results were obtained. Among these topological indices, one of most widely known topological description is the Wiener index, which was proposed in 1947 and represents the sum of the distances of all pairs of vertices in the graph, i.e.,. We describe the graphs with maximum RH index in terms of graph parameters such as vertex bipartiteness vb , (where 1 ≤ vb ≤ n − 2), cut edges, and matching numbers
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