On complexity and Jacobian of cone over a graph

TL;DR

This paper investigates the properties of a cone over a graph, specifically focusing on the complexity and Jacobian, providing formulas to compute these invariants using the graph's Laplacian.

Contribution

It establishes a direct relationship between the complexity and Jacobian of the cone over a graph and well-known graph invariants, offering explicit computational methods.

Findings

Complexity of the cone equals the number of rooted spanning forests in the original graph.

Jacobian of the cone is isomorphic to the cokernel of (I + Laplacian of G).

Complexity can be computed as the determinant of (I + Laplacian of G).

Abstract

For any given graph $G$ consider a graph $\widetilde{G}$ 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 $\widetilde{G}$ coincides the number of rooted spanning forests in graph $G$ and the Jacobian of $\widetilde{G}$ 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 $\widetilde{G}$ as $\det(I+L(G)).$