The critical group of a clique-inserted graph

In this paper, we consider the relations between the critical group of a regular graph G and that of its clique-inserted graph (or para-line graph) C ( G ) . First, we construct a group homomorphism between these two critical groups of G and C ( G ) . Based on the homomorphism, we show that the critical group of G is isomorphic to a quotient of that of C ( G ) if G is not bipartite, and the minimal number of generators for the critical group of C ( G ) is equal to the number of independent cycles in G if G is 2-edge connected. Second, by computing the Smith normal form of the Laplacian matrix of a graph, we obtain invariant factors of critical groups for some small regular graphs and their corresponding clique-inserted graphs.