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

TL;DR

This paper determines the critical groups of specific graph constructions involving complete and path graphs, expanding understanding of their algebraic properties in graph theory.

Contribution

It provides explicit calculations of the critical groups for the graphs formed by the join of complete and path graphs, a novel contribution in algebraic graph theory.

Findings

Critical groups of $K_m\vee P_n$ for $n\geq4$ are determined.

Critical groups of $P_m\vee P_n$ for $m\geq4$, $n\geq5$ are computed.

Results enhance understanding of algebraic invariants of graph joins.

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$$(n\geq4)$ and $P_m\sq P_n$$(m\geq4,n\geq5)$ are determined.