An Analog of Matrix Tree Theorem for Signless Laplacians

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.