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arXiv · 2002.03183

Proximity and remoteness in triangle-free and C_4-free graphs in terms of order and minimum degree

Peter Dankelmann·Elizabeth Jonck·Sonwabile Mafunda
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Abstract

Let $G$ be a finite, connected graph. The average distance of a vertex $v$ of $G$ is the arithmetic mean of the distances from $v$ to all other vertices of $G$. The remoteness $\rho(G)$ and the proximity $\pi(G)$ of $G$ are the maximum and the minimum of the average distances of the vertices of $G$. In this paper, we present a sharp upper bound on the remoteness of a triangle-free graph of given order and minimum degree, and a corresponding bound on the proximity, which is sharp apart from an additive constant. We also present upper bounds on the remoteness and proximity of $C_4$-free graphs of given order and minimum degree, and we demonstrate that these are close to being best possible.

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Peter Dankelmann, Elizabeth Jonck, Sonwabile Mafunda. 2020-02-08. Proximity and remoteness in triangle-free and C_4-free graphs in terms of order and minimum degree. https://arxiv.org/abs/2002.03183

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