A note on the critical ideals of a cycle

Abstract

Critical ideals of a graph G which are the determinantal ideals of its generalized Laplacian matrix were first introduced by Corrales and Valencia as a generalization of the critical group. Then it was shown that critical ideals are also closely related to other properties of the graph, such as the clique number and the zero forcing number. In this note, we give a simple proof of Theorem 4.13 proved in [7], which gives a Gröbner basis of the first nontrivial critical ideal of a cycle. After that as applications we determine explicit expressions for the characteristic ideals of a cycle and the critical groups of a family of thick wheels.