On the quotients between the eccentric connectivity index and the eccentric distance sum of graphs with diameter 2

For a connected graph G and a vertex v in G, let dG(v), εG(v) and DG(v) be the degree, eccentricity and distance sum of v, respectively. The eccentric connectivity index of G, denoted by ξc(G), is defined to be ξc(G)=∑v∈V(G)dG(v)εG(v), and the eccentric distance sum of G, denoted by ξd(G), is define...

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Bibliographic Details
Published in:Discrete Applied Mathematics 2020-10, Vol.285, p.297-300
Main Author: Hua, Hongbo
Format: Article
Language:English
Subjects:
Bounds
Connectivity
Eccentric connectivity index
Eccentric distance sum
Eccentricity
Extremal graphs
Graphs
Lower bounds
Quotient
Quotients
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Summary:For a connected graph G and a vertex v in G, let dG(v), εG(v) and DG(v) be the degree, eccentricity and distance sum of v, respectively. The eccentric connectivity index of G, denoted by ξc(G), is defined to be ξc(G)=∑v∈V(G)dG(v)εG(v), and the eccentric distance sum of G, denoted by ξd(G), is defined to be ξd(G)=∑v∈V(G)εG(v)DG(v). Denote by Gn2 the set of connected graphs of order n and diameter two. More recently, Zhang et al. (2019) investigated the relationship between the eccentric connectivity index and eccentric distance sum, and posed the problem to determine sharp upper and lower bounds on ξc(G)ξd(G) for graph in Gn2. In this short paper, we solve this problem. Sharp upper and lower bounds on ξc(G)ξd(G) for graph in Gn2 are determined, and the corresponding extremal graphs are characterized as well.
ISSN:0166-218X
1872-6771
DOI:10.1016/j.dam.2020.06.001