leavitt path algebras for order prime cayley graphs of finite groups

In this paper, we generalize the concept of cayley graphs associated to finite groups. the aim of this paper is the characterization of graph theoretic properties of new type of directed graph $gamma_p(g;s)$ and algebraic properties of leavitt path algebra of order prime cayley graph $ogamma(g;s)$, where $g$ is a finite group with a generating set $s$. we show that the leavitt path algebra of order prime cayley graph $l_k(ogamma(g;s))$ of a non trivial finite group $g$ with any generating set $s$ over a field $k$ is a purely infinite simple ring. finally, we prove that the grothendieck group of the leavitt path algebra $l_k(gamma_p(d_n;s))$ is isomorphic to $mathbb{z}_{2n-1}$, where $d_n$ is the dihedral group of degree $n$ and $s=left{a, bright}$ is the generating set of $d_n$.