Graphs with few paths of prescribed length between any two vertices

TL;DR

This paper constructs graphs with a specific number of edges and a bounded number of paths of a given length between any two vertices, using a novel algebraic method, and confirms the tightness of the edge bound.

Contribution

It introduces a new algebraic approach to construct graphs with controlled path counts, extending previous bounds on edges and path restrictions.

Findings

Graphs with (n^{1+1/k}) edges and bounded paths of length k exist

The construction uses a variant of Bukh's algebraic method

The edge bound is shown to be tight up to a constant

Abstract

We use a variant of Bukh's random algebraic method to show that for every natural number $k \geq 2$ there exists a natural number $\ell$ such that, for every $n$, there is a graph with $n$ vertices and $\Omega_k(n^{1 + 1/k})$ edges with at most $\ell$ paths of length $k$ between any two vertices. A result of Faudree and Simonovits shows that the bound on the number of edges is tight up to the implied constant.