An Analog of Matrix Tree Theorem for Signless Laplacians

TL;DR

This paper extends the Matrix Tree Theorem to signless Laplacians, providing a combinatorial interpretation for their principal minors and relating the determinant to the number of odd cycles in the graph.

Contribution

It introduces a novel combinatorial interpretation for principal minors of the signless Laplacian and establishes a bound on odd cycles related to its determinant.

Findings

Principal minors of the signless Laplacian have a combinatorial interpretation.

The number of odd cycles is bounded by the determinant of the signless Laplacian.

Equality in the bound characterizes bipartite and odd-unicyclic graphs.

Abstract

A spanning tree of a graph is a connected subgraph on all vertices with the minimum number of edges. The number of spanning trees in a graph $G$ is given by Matrix Tree Theorem in terms of principal minors of Laplacian matrix of $G$. We show a similar combinatorial interpretation for principal minors of signless Laplacian $Q$. We also prove that the number of odd cycles in $G$ is less than or equal to $\frac{\det(Q)}{4}$, where the equality holds if and only if $G$ is a bipartite graph or an odd-unicyclic graph.