On Mean Distance and Girth

TL;DR

This paper establishes bounds on the average distance in connected graphs based on their order and girth, providing tight inequalities that relate these parameters and characterizing cases of equality.

Contribution

It introduces new bounds on mean distance in graphs as a function of girth and order, extending previous understanding of graph metric properties.

Findings

Mean distance is at most rac{n+1}{3} minus a term depending on girth and order.

Mean distance is at least rac{ng}{4(n-1)} unless the graph is an odd cycle.

Bounds are tight and characterize extremal graphs.

Abstract

We bound the mean distance in a connected graph which is not a tree in function of its order $n$ and its girth $g$. On one hand, we show that mean distance is at most $\frac{n+1}{3}-\frac{g(g^2-4)}{12n(n-1)}$ if $g$ is even and at most $\frac{n+1}{3}-\frac{g(g^2-1)}{12n(n-1)}$ if $g$ is odd. On the other hand, we prove that mean distance is at least $\frac{ng}{4(n-1)}$ unless $G$ is an odd cycle.