Proximity in Triangulations and Quadrangulations

Czabarka, Éva; Dankelmann, Peter; Olsen, Trevor; Székely, László Á.

doi:10.48550/arxiv.2001.09012

“…Proximity of series-reduced trees, i.e., trees with no vertex of degree 2, were studied by Cheng, Lin and Zhou [5]. For maximal planar graphs, bounds on remoteness and proximity were given by Czabarka, Dankelmann, Olsen and Székely in [6] and [7], respectively. The study of proximity and remoteness in digraphs was initiated by Ai, Gerke, Gutin and Mafunda [1].…”

Section: Introductionmentioning

confidence: 99%

Proximity, remoteness and maximum degree in graphs

Dankelmann¹,

Mafunda²,

Mallu³

2022

Preprint

Self Cite

0

View full text Add to dashboard Cite

The average distance of a vertex v of a connected graph G is the arithmetic mean of the distances from v to all other vertices of G. The proximity π(G) and the remoteness ρ(G) of G are the minimum and the maximum of the average distances of the vertices of G, respectively.In this paper, we give upper bounds on the remoteness and proximity for graphs of given order, minimum degree and maximum degree. Our bounds are sharp apart from an additive constant.

show abstract

“…Proximity of series-reduced trees, i.e., trees with no vertex of degree 2, were studied by Cheng, Lin and Zhou [5]. For maximal planar graphs, bounds on remoteness and proximity were given by Czabarka, Dankelmann, Olsen and Székely in [6] and [7], respectively. The study of proximity and remoteness in digraphs was initiated by Ai, Gerke, Gutin and Mafunda [1].…”

Section: Introductionmentioning

confidence: 99%

Proximity, remoteness and maximum degree in graphs

Dankelmann¹,

Mafunda²,

Mallu³

2022

Discrete Mathematics &Amp; Theoretical Computer Science

0

View full text Add to dashboard Cite

The average distance of a vertex $v$ of a connected graph $G$ is the arithmetic mean of the distances from $v$ to all other vertices of $G$. The proximity $\pi(G)$ and the remoteness $\rho(G)$ of $G$ are the minimum and the maximum of the average distances of the vertices of $G$, respectively. In this paper, we give upper bounds on the remoteness and proximity for graphs of given order, minimum degree and maximum degree. Our bounds are sharp apart from an additive constant.

show abstract

“…Such bounds for trees had been obtained by Sedlar [17]). For further results on proximity and remoteness, see for example [1,3,6,7,8,9,10,15,18].…”

Section: Introductionmentioning

confidence: 99%

On the difference between proximity and other distance parameters in triangle-free graphs and $C_4$-free graphs

Dankelmann¹,

Mafunda²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

The average distance of a vertex v of a connected graph G is the arithmetic mean of the distances from v to all other vertices of G. The proximity π(G) and the remoteness ρ(G) of G are the minimum and the maximum of the average distances of the vertices of G. In this paper, we give upper bounds on the difference between the remoteness and proximity, the diameter and proximity, and the radius and proximity of a triangle-free graph with given order and minimum degree. We derive the latter two results by first proving lower bounds on the proximity in terms of order, minimum degree and either diameter or radius. Our bounds are sharp apart from an additive constant. We also obtain corresponding bounds for C 4 -free graphs.

show abstract

Proximity in Triangulations and Quadrangulations

Cited by 3 publications

References 15 publications

Proximity, remoteness and maximum degree in graphs

Proximity, remoteness and maximum degree in graphs

Proximity, remoteness and maximum degree in graphs

On the difference between proximity and other distance parameters in triangle-free graphs and $C_4$-free graphs

Contact Info

Product

Resources

About