IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 15, Issue 2 Ser. II (Mar – Apr 2019), PP 24-31 www.iosrjournals.org DOI: 10.9790/5728-1502022431 www.iosrjournals.org 24 | Page Circulant Graphs without Cayley Isomorphism Property with = V. Vilfred 1 and P. Wilson 2 Department of Mathematics 1 Central University of Kerala, Tejaswini HillsPeriye – 671 316, Kasaragod, Kerala, India. 2 S.T. Hindu College, Nagercoil – 629 002,Kanyakumari District, Tamil Nadu, India. Corresponding Author: V. Vilfred Abstract:A circulant graph () is said to have the Cayley Isomorphism (CI) property if whenever ()is isomorphic to (), there is some a∈ ∗ for which S = aR.In this paper, we prove that 27 (), 27 ()and 27 ()are isomorphic circulant graphs without CI-property whereR = {1, 9n-1, 9n+1, 3 1 , 3 2 , . . . , 3 −2 }, S = {3n+1, 6n-1, 12n+1, 3 1 , 3 2 , . . . , 3 −2 },T = {3n-1, 6n+1,12n-1, 3 1 , 3 2 , . . . ,3 −2 },k ≥ 3, gcd( 1 , 2 ,..., −2 ) = 1 and , 1 , 2 ,..., −2 ℕand also obtain new abelian groups from these isomorphic circulant graphs. ------------------------------------------------------------ AMS Subject Classification: 05C60, 05C25. Keywords:Adam's isomorphism or Type-1 isomorphism, Type-2 isomorphism, Cayley Isomorphism (CI) property, symmetric equidistance condition, Type-1 group on (),Type-2 group on () w.r.t. . --------------------------------------------------------------------------------------------------------------------------------------- Date of Submission: 27-03-2019Date of acceptance: 11-04-2019 ---------------------------------------------------------------------------------------------- I. Introduction Circulant graphs have been investigated by many authors [1]-[16]. An excellent account can be found in the book by Davis [3] and in [6]. A circulant graph () is said to have the Cayley Isomorphism (CI) property if whenever () is isomorphic to () there is some aℤ for which S = aR. Finding circulant graphs without CI-property is difficult. Type-2 isomorphism, a new type of isomorphism of circulant graphs, other than already known Adam's isomorphism, was defined and studied in [10,13]. Type-2 isomorphic circulant graphs have the property that they are isomorphic circulant graphs without CI-property. Families of isomorphic circulant graphs of Type-2, each circulant graph of a family with = gcd(n, ) number of copies of a circulant subgraph for = 2, 5 or 7 are obtained in [14]-[16].In this paper, we prove that for nℕ,k ≥ 3, R = {1, 9n-1, 9n+1, 3 1 , 3 2 , . . . , 3 −2 }, S = {3n+1, 6n-1, 12n+1, 3 1 , 3 2 , . . . , 3 −2 } andT = {3n-1, 6n+1,12n-1, 3 1 , 3 2 , ... , 3 −2 }, circulant graphs 27 (), 27 () and 27 () are Type-2 isomorphic with = 3 where gcd( 1 , 2 ,..., −2 ) = 1 and 1 , 2 ,..., −2 ℕ and obtain abelian groups( 27 ( 27 ()), o) = (1 27 ( 27 ()), o), ( 27,3 ( 27 ()), o) and (2 27 ,3 ( 27 ()), o). Through-out this paper, for a set R = { 1 , 2 , … , }, () denotes circulant graph ( 1 , 2 , … , ) where 1 1 < 2 < ⋯ < [n/2]. We consider only connected circulant graphs of finite order, V( ()) = { 0 , 1 , 2 , … , −1 } with adjacent to + for each rR, subscript addition taken modulo n and all cycles have length at least 3, unless otherwise specified,0 in-1. However when 2 R, edge + 2 is taken as a single edge for considering the degree of the vertex or + 2 and as a double edge while counting the number of edges or cycles in (), 0 in-1. Circulant graph is also defined as a Cayley graph or digraph of a cyclic group. If a graph G is circulant, then its adjacency matrix A(G) is circulant. It follows that if the first row of the adjacency matrix of a circulant graph is [ 1 , 2 , … , ], then 1 = 0 and = − +2 , 2 in [3]. We will often assume, with-out further comment, that the vertices are the corners of a regular n-gon, labeled clockwise. Circulant graphs 16 (1,2,7)and 16 (2,3,5) are shown in Figures 1 and 2, respectively. Now, we present a few definitions and results that are required in this paper. Theorem 1.1 [10]If () (), then there is a bijectionffromRtoSso that for allrR, gcd(n, r) = gcd(n, f(r)).