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

ANDREA L. GALLO; DENIS E. VIDELA

Discussiones Mathematicae Graph Theory

University of Zielona Gora

Año: 2024

1234-3099

In this work, given (R,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 GFq(k) a number |m| of times, with independence sets the cosets of m, where q is the size of the residue field R/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 GR/m(k) over the finite residue field R/m≃Fq. 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.