On a Conjecture of Randi\'{c} Index and Graph Radius
Hanyuan Deng, Zikai Tang, Jie Zhang

TL;DR
This paper investigates conjectures relating the Randić index and graph radius, proving new bounds using the harmonic index that improve previous results for various classes of graphs.
Contribution
The paper introduces bounds on the Randić index in terms of the graph radius using the harmonic index, strengthening existing conjectures for graphs with cyclomatic number and trees.
Findings
Proves R(G) ≥ r(G) - 31/105(k-1) for graphs with cyclomatic number k≥1.
Shows R(T) > r(T) + 1/15 for trees, excluding even paths.
Improves bounds on Randić index related to graph radius.
Abstract
The Randi\'{c} index of a graph is defined as the sum of over all edges of , where is the degree of the vertex in . The radius of a graph is the minimum graph eccentricity of any graph vertex in . Fajtlowicz(1988) conjectures for all connected graph . A stronger version, , is conjectured by Caporossi and Hansen(2000) for all connected graphs except even paths. In this paper, we make use of Harmonic index , which is defined as the sum of over all edges of , to show that for any graph with cyclomatic number , and for any tree except even paths. These results improve and strengthen the known results on these conjectures.
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Taxonomy
TopicsGraph theory and applications · Synthesis and Properties of Aromatic Compounds · Interconnection Networks and Systems
