The critical group of $K_m\vee P_n$ and $P_m\vee P_n$

Abstract

Let $G_1\sq G_2$ denote the graph obtained from $G_1+G_2$ by adding new edges from each vertex of $G_1$ to every vertex of $G_2$. In this paper, the critical groups of the graphs $K_m\sq P_n$$(n\ge 4)$ and $P_m\sq P_n$$(m\ge 4,n\ge 5)$ are determined.