Some alternating and symmetric groups and related graphs

Abstract

Let G be a finite group. The order-divisor graph $$\mathcal {S}^{*}(G)$$ as is the graph whose vertices are $$G{\setminus }\{1\}$$ and $$x\sim y$$ is an edge if and only if $$o(x)\mid o(y)$$ or $$o(y)\mid o(x)$$ . Let p be a prime number. In this paper, we will show that alternating group of degree $$p, p+1, p+2$$ and symmetric group of degree p are uniquely determined by their order-divisor graph.