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In this work, given $(R,\frak m)$ a finite commutative local ring withidentity and $k \in \mathbb{N}$ with $(k,|R|)=1$, we study the number ofcliques of any size in the Cayley graph $G_R(k)=Cay(R,U_R(k))$ %and$W_R(k)=Cay(R,S_R(k))$ with $U_R(k)=\{x^k : x\in R^*\}$. Using the known factthat the graph $G_R(k)$ can be obtained by blowing-up the vertices of$G_{\mathbb{F}_{q}}(k)$ a number $|\frak{m}|$ of times, with independence setsthe cosets of $\frak{m}$, where $q$ is the size of the residue field $R/\frakm$. Then, by using the above blowing-up, we reduce the study of the number ofcliques in $G_R(k)$ over the local ring $R$ to the computation of the number ofcliques of $G_{R/\frak{m}}(k)$ over the finite residue field $R/\frak m \simeq\mathbb{F}_q$. In this way, using known numbers of cliques of generalized Paleygraphs ($k=2,3,4$ and $\ell=3,4$), we obtain several explicit results for thenumber of cliques over finite commutative local rings with identity.

Number of cliques of Paley-type graphs over finite commutative local rings