2020/12/18 by V. Vilfred Kamalappan, Kamalappan, V. Vilfred · 1 citation
Mathematics · Computer Science · #Finite Group Theory Research #Graph theory and applications #Interconnection Networks and Systems
paper · pdf · doi:10.48550/arxiv.2012.11372
Circulant graphs Cn(R) and Cn(S) are said to be Adam's isomorphic if there exist some a∈ ℤn^* such that S = a R under arithmetic reflexive modulo n. In 1970, Elspas and Turner \citeeltu raised a question on the isomorphism of C16(1, 3, 7) and C16(2, 3, 5) and Vilfred \citev96 gave its answer by defining Type-2 isomorphism, different from Adam's isomorphism or Type-1 isomorphism, of Cn(R) w.r.t. m where m > 1 is a divisor of gcd(n, r) and r∈ R. This paper is an extensive study on Type-2 isomorphic circulant graphs. Vilfred and Wilson \citevw0A obtain isomorphic circulant graphs Cnp3(R) of Type-2 w.r.t. m = p, and related Abelian groups where p is a prime number and n∈ℕ. Using Theorem \refc13, a list of T2np3,p(Cnp3(Rnp3,x+ypi)) = \Cnp3(Rnp3,x+ypj) : j = 1,2,...,p\ for p = 3,5,7,11 and n = 1 to 5 and also for p = 13 and n = 1 to 3 are given in the Annexure where (T2np3,p(Cnp3(Rnp3,x+ypi)), ∘) is an abelian group on the p isomorphic circulant graphs Cnp3(Rnp3,x+ypi) of Type-2 w.r.t. m = p, 1 ≤ i,j ≤ p, 1 ≤ x ≤ p-1, y∈ℕ0, 0 ≤ y ≤ np - 1, 1 ≤ x+yp ≤ np2-1, p,np3-p∈ Rnp3,x+ypi and i,j,n,x∈ℕ. We also show existence of isomorphic circulant graphs Cn(R) and Cn(S) which are neither Type-1 nor Type-2 w.r.t. any particular m. We use VB program to develop this theory and for illustration of examples.