The Number of Spanning Trees in 4-Regular Simple Graphs

Abstract

Extending an earlier work by Kostochka for subcubic graphs, we show that a connected graph $G$ with minimum degree $2$ and maximum degree $4$ has at least $75^{n_4/5+n_3/10+1/5}$ spanning trees, where $n_i$ is the number of vertices of degree $i$ in $G$, unless $G$ is the complete graph on $5$ vertices or obtained from the complete graph on $6$ vertices by deleting the edges of a perfect matching. This, in particular, allows us to determine the value of the inferior limit of the normalised number of spanning trees (introduced by Alon) over the class of connected $4$-regular graphs to be $75^{1/5}$.