  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.
