Consider a finite group \(G\), a subset \(S\subset G\) with \(S\) any nonempty generating set of \(G\), a map \(w\colon S \to {\mathbb N}\) and \(\hbox{Cay}(G,S,w)\) the weighted Cayley graph. In the particular case that \(n\) is a positive integer and \(G={\mathbb Z}_n\) one can denote \(C_n(S,w)=\hbox{Cay}({\mathbb Z}_n,S,w)\). In Theorem 3.1 of this paper, the author finds that \(L(\hbox{Cay}(G,S,w))\) is a purely infinite simple Leavitt path algebra if and only if \(W:=\sum_{s \in S}w(s) \ge 2\) and if and only if \(L(\hbox{Cay}(G,S,w))\) does not have the IBN property. In the case \(S=\{s_1,s_2, \ldots , s_k\} \subsetneq {\mathbb Z}_n, \ s_1<s_2< \cdots <s_k\), \(0 \notin S\) and \(\sum_{s_j \in S}w(s_j) \ge 2\) (\(L(C_n(S,w))\) a purely infinite simple Leavitt path algebra), the author shows how to calculate its Grothendieck group. Also other specific Leavitt path algebras of Cayley graphs and its Grothendieck group \(K_0(L(C_n(S,w)))\) are described. So is the case of Leavitt path algebras associated to Cayley graphs for dihedral groups.