Abstract
The Resistance-Harary index of a connected graph G is defined as R H ( G ) = ∑ { u , v } ⊆ V ( G ) 1 r ( u , v ) , where r ( u , v ) is the resistance distance between vertices u and v in G. A graph G is called a unicyclic graph if it contains exactly one cycle and a fully loaded unicyclic graph is a unicyclic graph that no vertex with degree less than three in its unique cycle. Let U ( n ) and U ( n ) be the set of unicyclic graphs and fully loaded unicyclic graphs of order n, respectively. In this paper, we determine the graphs of U ( n ) with second-largest Resistance-Harary index and determine the graphs of U ( n ) with largest Resistance-Harary index.
Highlights
The topological index is the mathematical descriptor of the molecular structure, which can effectively reflect the chemical structure and properties of the material
In the chemical graph representing the non-hydrogen atoms in the molecule.In 1993, Klein and Randić [1] defined a new distance function named resistance distance on the basis of electrical network theory replacing each edge of a simple connected graph G by a unit resistor
If the ordinary distance is replaced by resistance distance in the expression for the Wiener index, one arrives at the Kirchhoff index [1,2]
Summary
The topological index is the mathematical descriptor of the molecular structure, which can effectively reflect the chemical structure and properties of the material. R (u, v), which has been widely studied [3,4,5,6,7,8,9,10,11,12] Another distance-based graph invariant index named Harary index was introduced independently by Plavšić et al [13] and by Ivanciuc et al [14] in 1993 for the characterization of molecular graphs. In 2017, Chen et al [23,24] introduced a new graph invariant reciprocal to Kirchhoff index, named Resistance-Harary index, as RH ( G ) =. Resistance-Harary index among all fully loaded unicyclic graphs and characterize the corresponding extremal graphs, respectively
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