Document Type
Article
Publication Date
11-2022
Abstract
For a simple signed graph G with the adjacency matrix A and net degree matrix D±, the net Laplacian matrix is L± = D±−A. We introduce a new oriented incidence matrix N± which can keep track of the sign as well as the orientation of each edge of G. Also L± = N±(N±)T. Using this decomposition, we find the number of both positive and negative spanning trees of G in terms of the principal minors of L± generalizing the Matrix Tree Theorem for an unsigned graph. We present similar results for the signless net Laplacian matrix Q± = D± +A along with a combinatorial formula for its determinant.
Recommended Citation
Mallik, S. (2024). Matrix tree theorem for the net Laplacian matrix of a signed graph. Linear and Multilinear Algebra, 72(7), 1138–1152. https://doi.org/10.1080/03081087.2023.2172544
Comments
This is the author’s original manuscript that was submitted to arXiv. The version of record is available from the publisher at https://doi.org/10.1080/03081087.2023.2172544