Quadratic Residue Cayley Graphs on Composite Modulus

Abstract

In this paper, we initiate the study of quadratic residue Cayley graphs $\varGamma _N$ modulo $N=pq$, where p, q are distinct primes of the form $4k+1$. It is shown that $\varGamma _N$ is a regular, symmetric, Eulerian, and Hamiltonian graph. Also, the vertex connectivity, edge connectivity, diameter, and girth of $\varGamma _N$ are studied and their relationship with the forms of p and q are discussed. Moreover, we specify the forms of primes for which $\varGamma _N$ is triangulated or triangle-free and provide some bounds for the order of the automorphism group of $\varGamma _N$, $Aut(\varGamma _N)$ and domination number of $\varGamma _N$.

The author’s research is supported in part by National Board of Higher Mathematics, Department of Atomic Energy, Government of India (No 2/48(10)/2013/NBHM(R.P.)/R&D II/695).