A formula for the number of the spanning trees of line graphs

TL;DR

This paper derives a new formula for counting spanning trees in line graphs of general graphs using electrical network techniques, simplifying proofs and confirming a previous conjecture.

Contribution

It introduces a novel formula for the number of spanning trees in line graphs, providing simpler proofs and confirming a conjecture by Yan.

Findings

Derived a new formula for spanning trees of line graphs.

Confirmed Yan's conjecture on irregular line graphs.

Applied the formula to circulant line graphs.

Abstract

Let $G=(V,E)$ be a loopless graph and $\mathcal{T}(G)$ be the set of all spanning trees of $G$. Let $L(G)$ be the line graph of the graph $G$ and $t(L(G))$ be the number of spanning trees of $L(G)$. Then, by using techniques from electrical networks, we obtain the following formula: $$ t(L(G)) = \frac{1}{\prod_{v\in V}d^2(v)}\sum_{T\subseteq \mathcal{T}(G)}\big[\prod_{e = xy\in T}d(x)d(y)\big]\big[\prod_{e = uv\in E\backslash T}[d(u)+d(v)]\big]. $$ As a result, we provide a very simple and different proof of the formula on the number of spanning trees of some irregular line graphs, and give a positive answer to a conjecture proposed by Yan [J. Combin. Theory Ser. A 120 (2013) no. 7, 1642-1648]. By applying our formula we also derive the number of spanning trees of circulant line graphs.