2023/03/08 by Gallo, Andrea L., Videla, Denis E.
#05C25 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2303.04312
In this work, given (R,\frak m) a finite commutative local ring with identity and k ∈ ℕ with (k,|R|)=1, we study the number of cliques of any size in the Cayley graph GR(k)=Cay(R,UR(k)) %and WR(k)=Cay(R,SR(k)) with UR(k)=\xk : x∈ R^*\. Using the known fact that the graph GR(k) can be obtained by blowing-up the vertices of G_\mathbbFq(k) a number |\frakm| of times, with independence sets the cosets of \frakm, where q is the size of the residue field R/\frak m. Then, by using the above blowing-up, we reduce the study of the number of cliques in GR(k) over the local ring R to the computation of the number of cliques of G_R/\frakm(k) over the finite residue field R/\frak m ≃ \mathbbFq. In this way, using known numbers of cliques of generalized Paley graphs (k=2,3,4 and ℓ=3,4), we obtain several explicit results for the number of cliques over finite commutative local rings with identity.