On the eccentric connectivity index and Wiener index of a graph

P. Dankelmann; M. J. Morgan; S. Mukwembi; H. C. Swart

doi:10.2989/16073606.2013.779963

On the eccentric connectivity index and Wiener index of a graph

P. Dankelmann
, M. J. Morgan
, S. Mukwembi
, H. C. Swart

Mathematics and Applied Mathematics

University of KwaZulu-Natal

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

21 Citations (Scopus)

Abstract

Let G be a finite connected graph of order n and minimum degree δ. The eccentric connectivity index ξ^c (G) of G is defined as ξ^c (G) = Σ_{v{small element of}V (G)} ec_G (v)deg_G (v), where ec_G (x) and deg_G (x) denote the eccentricity and degree of vertex x in G, respectively. We prove that the eccentric connectivity index of G satisfies, and construct graphs which asymptotically attain the bound. Our bound implies some known results by Došlić, Saheli & Vukičević [4], Morgan, Mukwembi & Swart [11], and Zhou & Du [16]. Further, we also determine upper bounds on the well-studied Wiener index in terms of the eccentric connectivity index.

Original language	English
Pages (from-to)	39-47
Number of pages	9
Journal	Quaestiones Mathematicae
Volume	37
Issue number	1
DOIs	https://doi.org/10.2989/16073606.2013.779963
Publication status	Published - Jan 2014

Keywords

Eccentricity
eccentric connectivity index
minimum degree

ASJC Scopus subject areas

Mathematics (miscellaneous)

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10.2989/16073606.2013.779963

Cite this

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AU - Dankelmann, P.

AU - Morgan, M. J.

AU - Mukwembi, S.

AU - Swart, H. C.

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N2 - Let G be a finite connected graph of order n and minimum degree δ. The eccentric connectivity index ξc (G) of G is defined as ξc (G) = Σv{small element of}V (G) ecG (v)degG (v), where ecG (x) and degG (x) denote the eccentricity and degree of vertex x in G, respectively. We prove that the eccentric connectivity index of G satisfies, and construct graphs which asymptotically attain the bound. Our bound implies some known results by Došlić, Saheli & Vukičević [4], Morgan, Mukwembi & Swart [11], and Zhou & Du [16]. Further, we also determine upper bounds on the well-studied Wiener index in terms of the eccentric connectivity index.

AB - Let G be a finite connected graph of order n and minimum degree δ. The eccentric connectivity index ξc (G) of G is defined as ξc (G) = Σv{small element of}V (G) ecG (v)degG (v), where ecG (x) and degG (x) denote the eccentricity and degree of vertex x in G, respectively. We prove that the eccentric connectivity index of G satisfies, and construct graphs which asymptotically attain the bound. Our bound implies some known results by Došlić, Saheli & Vukičević [4], Morgan, Mukwembi & Swart [11], and Zhou & Du [16]. Further, we also determine upper bounds on the well-studied Wiener index in terms of the eccentric connectivity index.

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On the eccentric connectivity index and Wiener index of a graph

Abstract

Keywords

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