Matrix tree theorem for the net Laplacian matrix of a signed graph (2210.03711v2)
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.
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