Enumeration of Triangles in Cayley Graphs

Abstract

Significant contributions can be found on the study of the cycle structure in graphs, particularly in Cayley graphs.Determination of Hamilton cycles and triangles, the longest and shortest cycles attracts special attention. In this paper an enumeration process for the determination of number of triangles in the Cayley graph associated with a group not necessarily abelian and a symmetric subset of the group.

Unitary Cayley graphs of Dedekind domain quotients

“…Observe that if we set m = 3 in Corollary 3.1, we recover the previously-discovered formula 1 6 nφ(n)S 2 (n) for the number of triangles in G Z/(n) . Our proof of this formula seems much more natural and illuminating than those already in existence [1,2,3]. Before proceeding to uncover some additional properties of generalized totient graphs, we pause to note an interesting divisibility relationship that arises as a corollary of Theorem 3.1.…”

“…If X is a commutative ring with unity, then the unitary Cayley graph of X, denoted G X , is defined to be the graph whose vertex set is X and whose edge set is {{a, b} : a − b ∈ X × }. The unitary Cayley graphs G Z/(n) for n ∈ Z + have been named "Euler totient Cayley graphs" [3,4,5]. Several researchers have shown that the graph G Z/(n) contains exactly 1 6 nφ(n)S 2 (n) triangles [1,2,3], and Manjuri and Maheswari have studied Euler totient Cayley graphs in the context of domination parameters [4,5].…”

“…The unitary Cayley graphs G Z/(n) for n ∈ Z + have been named "Euler totient Cayley graphs" [3,4,5]. Several researchers have shown that the graph G Z/(n) contains exactly 1 6 nφ(n)S 2 (n) triangles [1,2,3], and Manjuri and Maheswari have studied Euler totient Cayley graphs in the context of domination parameters [4,5]. Klotz and Sander have studied, among other properties, the diameters and eigenvalues of Euler totient Cayley graphs [2], and their paper gives a list of references to other results related to these graphs.…”

Enumeration of disjoint Hamilton cycles in a divisor Cayley graph

“…The following corollary is immediate from the Theorem 3.3. Step 1: For the divisor d = 5, the 5 disjoint cycles of length 3 are (0, 5, 10, 0), (1,6,11,1) , (2,7,12,2) , (3,8,13,3) and (4,9,14,4).…”

Hamiltonian property of intersection graph of zero divisors of the ring Zn

“…Generally, the Hamilton problem is considered to be determining the conditions under which a graph contains a spanning cycle. Many authors have studied the Hamilton cycles for several types of graphs and in which those are refer [6,13,14].…”