Remoteness of graphs with given size and connectivity constraints

TL;DR

This paper establishes precise upper bounds on the remoteness of graphs based on their size, order, and connectivity, including special cases for triangle-free and edge-connected graphs.

Contribution

It provides new sharp bounds on graph remoteness considering various connectivity and structural constraints, extending previous results.

Findings

Sharp upper bounds for remoteness based on order, size, and connectivity.

Bounds specifically for 2-edge-connected and 3-edge-connected graphs.

Results for triangle-free graphs in terms of order and size.

Abstract

Let $G$ be a finite, simple 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 $\rho(G)$ of $G$ is the maximum of the average distances of the vertices of $G$. In this paper, we give sharp upper bounds on the remoteness of a graph of given order, connectivity and size. We also obtain corresponding bound s for $2$-edge-connected and $3$-edge-connected graphs, and bounds in terms of order and size for triangle-free graphs.