Average Distance and Edge-Connectivity II

Dankelmann, Peter; Mukwembi, Simon; Swart, Henda C.

doi:10.1137/06065653x

Cited by 16 publications

(17 citation statements)

References 4 publications

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“…Define a bijection f : which is (10). By the construction of T 0 and T every edge b i , i > 1, is at distance exactly three in T from some edge b j with j < i.…”

Section: An Improved Bound For Triangle-free Graphsmentioning

confidence: 99%

The Steiner k-Wiener index of graphs with given minimum degree

Dankelmann

2019

Discrete Applied Mathematics

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“…Define a bijection f : which is (10). By the construction of T 0 and T every edge b i , i > 1, is at distance exactly three in T from some edge b j with j < i.…”

Section: An Improved Bound For Triangle-free Graphsmentioning

confidence: 99%

The Steiner k-Wiener index of graphs with given minimum degree

Dankelmann

2019

Discrete Applied Mathematics

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“…Plesník [12] posed the problem of finding sharp upper bounds on (G) for k-vertex-connected (k-edge-connected) graphs, where k ≥ 3. The present authors [2,3] established asymptotically sharp upper bounds on the average distance of a k-edgeconnected graph of given order, where k ≥ 3. Favaron et al [8] generalized Plesník's bound and proved that if G is a -vertex-connected graph of order n, then (G) ≤ n + −1 n −1− 2 n −1…”

Section: Introductionmentioning

confidence: 98%

Average distance and vertex‐connectivity

Dankelmann

Mukwembi

Swart

2009

Journal of Graph Theory

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Abstract:The average distance (G) of a connected graph G of order n is the average of the distances between all pairs of vertices of G, i.e., y), where V(G) denotes the vertex set of G and d G (x, y) is the distance between x and y. We prove that if G is a -vertex-connected graph, ≥ 3 an odd integer, of order n, then (G) ≤ n/2( +1)+O(1). Our bound is shown to be best possible and our results, apart from answering a question of Plesník [J Graph Theory 8 (1984), 1--24], Favaron et al. [Networks 19 (1989), 493--504], can be used to deduce a theorem that is essentially equivalent to a theorem by Egawa and Inoue [Ars Combin 51 (1999), 89--95] on the radius of a -vertex-connected graph of given order, where is odd. ᭧

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“…which is attained only by a path (Dankelmann et al 2008a;Plesník 1984;Lovász 1979). Many sharp or asymptotically sharp bounds on W (G) in terms of other graph parameters are known, for instance, minimum degree (Beezer et al 2001;Dankelmann and Entringer 2000;Kouider and Winkler 1997), connectivity (Dankelmann et al 2009;Favaron et al 1989), edge-connectivity (Dankelmann et al 2008b, a) and maximum degree (Fischermann et al 2002). For finding more details in mathematical aspect of Wiener index, see also results (Das and Nadjafi-Arani 2017;Gutman et al 2014;Klavžar and Nadjafi-Arani 2014;Knor et al , 2016Li et al 2018;Mukwembi and Vetrík 2014;Wagner et al 2009;Wagner 2006;Wang and Yu 2006).…”

Section: Introductionmentioning

confidence: 99%

The maximum Wiener index of maximal planar graphs

Ghosh¹,

Győri

Paulos

et al. 2020

J Comb Optim

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The Wiener index of a connected graph is the sum of the distances between all pairs of vertices in the graph. It was conjectured that the Wiener index of an n-vertex maximal planar graph is at most $$\lfloor \frac{1}{18}(n^3+3n^2)\rfloor $$ ⌊ 1 18 ( n 3 + 3 n 2 ) ⌋ . We prove this conjecture and determine the unique n-vertex maximal planar graph attaining this maximum, for every $$ n\ge 10$$ n ≥ 10 .

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Average Distance and Edge-Connectivity II

Cited by 16 publications

References 4 publications

The Steiner k-Wiener index of graphs with given minimum degree

The Steiner k-Wiener index of graphs with given minimum degree

Average distance and vertex‐connectivity

The maximum Wiener index of maximal planar graphs

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