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On a conjecture of randić index and graph radius

On a conjecture of randić index and graph radius
On a conjecture of randić index and graph radius
The Randić index R(G) of a graph G is defined as the sum of (didj)- (formula presented) over all edges vivj of G, where di is the degree of the vertex vi in G. The radius r(G) of a graph G is the minimum graph eccentricity of any graph vertex in G. Fajtlowicz in [S. Fajtlowicz, On conjectures of Graffiti, Discrete Math. 72 (1988) 113-118] conjectures R(G) ≥ r(G) - 1 for any connected graph G. A stronger version, R(G) ≥ r(G), is conjectured for all connected graphs except even paths by Caporossi and Hansen in [G. Caporossi, et al., Variable neighborhood search for extremal graphs 1: The Autographix system, Discrete Math. 212 (2000) 29-44]. In this paper, we make use of Harmonic index H(G), which is defined as the sum of (formula presented) over all edges vivj of G, to show that R(G) ≥ r(G) - (formula presented) (k - 1) for any graph with cyclomatic number k ≥ 1, and R(T) > r(T) + (formula presented) for any tree except even paths. These results improve and strengthen the known results on these conjectures.
Graph, Harmonic index, Radius, Randi´c index
0354-5180
1369-1375
Denga, Hanyuan
3f4a0c71-89de-416a-92e2-db6ac54b4734
Tanga, Zikai
9bce7d3b-38f8-433b-ae03-386cb5d1a1e0
Zhang, Jie
6bad4e75-40e0-4ea3-866d-58c8018b225a
Denga, Hanyuan
3f4a0c71-89de-416a-92e2-db6ac54b4734
Tanga, Zikai
9bce7d3b-38f8-433b-ae03-386cb5d1a1e0
Zhang, Jie
6bad4e75-40e0-4ea3-866d-58c8018b225a

Denga, Hanyuan, Tanga, Zikai and Zhang, Jie (2015) On a conjecture of randić index and graph radius. Filomat, 29 (6), 1369-1375. (doi:10.2298/FIL1506369D).

Record type: Article

Abstract

The Randić index R(G) of a graph G is defined as the sum of (didj)- (formula presented) over all edges vivj of G, where di is the degree of the vertex vi in G. The radius r(G) of a graph G is the minimum graph eccentricity of any graph vertex in G. Fajtlowicz in [S. Fajtlowicz, On conjectures of Graffiti, Discrete Math. 72 (1988) 113-118] conjectures R(G) ≥ r(G) - 1 for any connected graph G. A stronger version, R(G) ≥ r(G), is conjectured for all connected graphs except even paths by Caporossi and Hansen in [G. Caporossi, et al., Variable neighborhood search for extremal graphs 1: The Autographix system, Discrete Math. 212 (2000) 29-44]. In this paper, we make use of Harmonic index H(G), which is defined as the sum of (formula presented) over all edges vivj of G, to show that R(G) ≥ r(G) - (formula presented) (k - 1) for any graph with cyclomatic number k ≥ 1, and R(T) > r(T) + (formula presented) for any tree except even paths. These results improve and strengthen the known results on these conjectures.

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More information

Published date: 2015
Additional Information: Publisher Copyright: © 2015 University of Nis. All Rights Reserved.
Keywords: Graph, Harmonic index, Radius, Randi´c index

Identifiers

Local EPrints ID: 474786
URI: http://eprints.soton.ac.uk/id/eprint/474786
ISSN: 0354-5180
PURE UUID: 04be27a1-99a2-4632-8800-c1604697d6c2

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Date deposited: 02 Mar 2023 17:48
Last modified: 05 Jun 2024 19:57

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Contributors

Author: Hanyuan Denga
Author: Zikai Tanga
Author: Jie Zhang

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