Spanning Trees of Bounded Degree Graphs

TL;DR

This paper investigates lower bounds on the number of spanning trees in connected graphs with bounded degree, aiming to improve exponential algorithm analysis, and conjectures the extremal case is achieved by complete graphs.

Contribution

It introduces a conjecture on the minimal number of spanning trees in bounded degree graphs and provides new lower bounds for degrees up to 11.

Findings

Conjecture that complete graphs maximize the number of spanning trees for given degree bounds.

Established weaker lower bounds on the number of spanning trees for degrees up to 11.

Potential implications for analyzing exponential algorithms with memorization.

Abstract

We consider lower bounds on the number of spanning trees of connected graphs with degree bounded by $d$. The question is of interest because such bounds may improve the analysis of the improvement produced by memorisation in the runtime of exponential algorithms. The value of interest is the constant $\beta_d$ such that all connected graphs with degree bounded by $d$ have at least $\beta_d^\mu$ spanning trees where $\mu$ is the cyclomatic number or excess of the graph, namely $m-n+1$. We conjecture that $\beta_d$ is achieved by the complete graph $K_{d+1}$ but we have not proved this for any $d$ greater than 3. We give weaker lower bounds on $\beta_d$ for $d\le 11$.