Matrix tree theorem for the net Laplacian matrix of a signed graph

TL;DR

This paper extends the Matrix Tree Theorem to signed graphs by introducing a new incidence matrix and expressing the counts of positive and negative spanning trees through principal minors of the net Laplacian.

Contribution

It introduces a novel oriented incidence matrix for signed graphs and generalizes the Matrix Tree Theorem to account for positive and negative spanning trees.

Findings

Derived formulas for counting positive and negative spanning trees.

Established a relationship between the net Laplacian and spanning trees.

Provided a combinatorial formula for the determinant of the signless net Laplacian.

Abstract

For a simple signed graph $G$ with the adjacency matrix $A$ and net degree matrix $D^{\pm}$, the net Laplacian matrix is $L^{\pm}=D^{\pm}-A$. We introduce a new oriented incidence matrix $N^{\pm}$ which can keep track of the sign as well as the orientation of each edge of $G$. Also $L^{\pm}=N^{\pm}(N^{\pm})^T$. Using this decomposition, we find the numbers of positive and negative spanning trees of $G$ in terms of the principal minors of $L^{\pm}$ generalizing Matrix Tree Theorem for an unsigned graph. We present similar results for the signless net Laplacian matrix $Q^{\pm}=D^{\pm}+A$ along with a combinatorial formula for its determinant.