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

Ke Xiang Xu; Kinkar Chandra Das; Xiao Qian Gu

doi:10.1007/s10114-019-8439-9

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

Ke Xiang Xu, Kinkar Chandra Das, Xiao Qian Gu

Department of Mathematics

Nanjing University of Aeronautics and Astronautics

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

Abstract

The first and second Zagreb eccentricity indices of graph G are defined as: E1(G)=∑vi∈V(G)εG(vi)2, E2(G)=∑vivj∈E(G)εG(vi)εG(vj) 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 E1(G)n and E2(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.

Original language	English
Pages (from-to)	40-54
Number of pages	15
Journal	Acta Mathematica Sinica, English Series
Volume	36
Issue number	1
DOIs	https://doi.org/10.1007/s10114-019-8439-9
State	Published - 1 Jan 2020

Keywords

05C12
05C35
diameter
eccentric complexity
Eccentricity (of vertex)
first Zagreb eccentricity index
second Zagreb eccentricity index

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10.1007/s10114-019-8439-9

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title = "Comparison and Extremal Results on Three Eccentricity-based Invariants of Graphs",

abstract = "The first and second Zagreb eccentricity indices of graph G are defined as: E1(G)=∑vi∈V(G)εG(vi)2, E2(G)=∑vivj∈E(G)εG(vi)εG(vj) where εG(υi) denotes the eccentricity of vertex υi in G. The eccentric complexity Cec(G) of G is the number of different eccentricities of vertices in G. In this paper we present some results on the comparison between E1(G)n and E2(G)m for any connected graphs G of order n with m edges, including general graphs and the graphs with given Cec. Moreover, a Nordhaus-Gaddum type result Cec(G) + Cec(Ḡ) is determined with extremal graphs at which the upper and lower bounds are attained respectively.",

keywords = "05C12, 05C35, diameter, eccentric complexity, Eccentricity (of vertex), first Zagreb eccentricity index, second Zagreb eccentricity index",

author = "Xu, \{Ke Xiang\} and Das, \{Kinkar Chandra\} and Gu, \{Xiao Qian\}",

note = "Publisher Copyright: {\textcopyright} 2019, Springer-Verlag GmbH Germany \& The Editorial Office of AMS.",

year = "2020",

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doi = "10.1007/s10114-019-8439-9",

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TY - JOUR

T1 - Comparison and Extremal Results on Three Eccentricity-based Invariants of Graphs

AU - Xu, Ke Xiang

AU - Das, Kinkar Chandra

AU - Gu, Xiao Qian

PY - 2020/1/1

Y1 - 2020/1/1

N2 - The first and second Zagreb eccentricity indices of graph G are defined as: E1(G)=∑vi∈V(G)εG(vi)2, E2(G)=∑vivj∈E(G)εG(vi)εG(vj) where εG(υi) denotes the eccentricity of vertex υi in G. The eccentric complexity Cec(G) of G is the number of different eccentricities of vertices in G. In this paper we present some results on the comparison between E1(G)n and E2(G)m for any connected graphs G of order n with m edges, including general graphs and the graphs with given Cec. Moreover, a Nordhaus-Gaddum type result Cec(G) + Cec(Ḡ) is determined with extremal graphs at which the upper and lower bounds are attained respectively.

AB - The first and second Zagreb eccentricity indices of graph G are defined as: E1(G)=∑vi∈V(G)εG(vi)2, E2(G)=∑vivj∈E(G)εG(vi)εG(vj) where εG(υi) denotes the eccentricity of vertex υi in G. The eccentric complexity Cec(G) of G is the number of different eccentricities of vertices in G. In this paper we present some results on the comparison between E1(G)n and E2(G)m for any connected graphs G of order n with m edges, including general graphs and the graphs with given Cec. Moreover, a Nordhaus-Gaddum type result Cec(G) + Cec(Ḡ) is determined with extremal graphs at which the upper and lower bounds are attained respectively.

KW - 05C12

KW - 05C35

KW - diameter

KW - eccentric complexity

KW - Eccentricity (of vertex)

KW - first Zagreb eccentricity index

KW - second Zagreb eccentricity index

UR - https://www.scopus.com/pages/publications/85077569777

U2 - 10.1007/s10114-019-8439-9

DO - 10.1007/s10114-019-8439-9

M3 - Article

AN - SCOPUS:85077569777

SN - 1439-8516

VL - 36

SP - 40

EP - 54

JO - Acta Mathematica Sinica, English Series

JF - Acta Mathematica Sinica, English Series

IS - 1

ER -

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

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Keywords

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