Leavitt path algebras of weighted Cayley graphs $C_n(S,w)$

For a postive integer $n$ and a subset $S$ of $\mathbb{Z}_n$, let $\left\langle S\right\rangle =\mathbb{Z}_n$, and $w:S\rightarrow\mathbb{N}$ be a function. The weighted Cayley graph of the cyclic group $\mathbb{Z}_n$ with respect to $S$ and $w$ is denoted by $C_n(S,w)$. We give an explicit description of the Grothendieck group of the Leavitt path algebras of $C_n(S,w)$. We also give description of Leavitt path algebras of $C_n(S,w)$ in some special cases.