Comparison and Extremal Results on Three Eccentricity-based Invariants of Graphs

The first and second Zagreb eccentricity indices of graph G are defined as: E 1 ( G ) = ∑ v i ∈ V ( G ) ε G ( v i ) 2 , E 2 ( G ) = ∑ v i v j ∈ E ( G ) ε G ( v i ) ε G ( v j ) where ε G ( υ i ) denotes the eccentricity of vertex υ i in G . The eccentric complexity C ec ( G ) of G is the number of di...

Full description

Saved in:
Bibliographic Details
Published in:Acta mathematica Sinica. English series 2020, Vol.36 (1), p.40-54
Main Authors: Xu, Ke Xiang, Das, Kinkar Chandra, Gu, Xiao Qian
Format: Article
Language:English
Subjects:
Apexes
Eccentricity
Graphs
Lower bounds
Mathematical analysis
Mathematics
Mathematics and Statistics
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:The first and second Zagreb eccentricity indices of graph G are defined as: E 1 ( G ) = ∑ v i ∈ V ( G ) ε G ( v i ) 2 , E 2 ( G ) = ∑ v i v j ∈ E ( G ) ε G ( v i ) ε G ( v j ) where ε G ( υ i ) denotes the eccentricity of vertex υ i in G . The eccentric complexity C ec ( G ) of G is the number of different eccentricities of vertices in G . In this paper we present some results on the comparison between E 1 ( G ) n and E 2 ( G ) m for any connected graphs G of order n with m edges, including general graphs and the graphs with given C ec . Moreover, a Nordhaus-Gaddum type result C ec ( G ) + C ec ( Ḡ ) is determined with extremal graphs at which the upper and lower bounds are attained respectively.
ISSN:1439-8516
1439-7617
DOI:10.1007/s10114-019-8439-9