On complexity and Jacobian of cone over a graph [article]

L. A. Grunwald, I.A. Mednykh
2020 arXiv   pre-print
For any given graph G consider a graph G which is a cone over graph G. In this paper, we study two important invariants of such a cone. Namely, complexity (the number of spanning trees) and the Jacobian of a graph. We prove that complexity of graph G coincides the number of rooted spanning forests in graph G and the Jacobian of G is isomorphic to cokernel of the operator I+L(G), where L(G) is Laplacian of G and I is the identity matrix. As a consequence, one can calculate the complexity of G as (I+L(G)).
arXiv:2004.07452v1 fatcat:qwddgrouevdgpjvyphlayv4iva