Some Properties of Unitary Cayley Graphs

Abstract

The unitary Cayley graph $X_n$ has vertex set $Z_n=\{0,1, \ldots ,n-1\}$. Vertices $a, b$ are adjacent, if gcd$(a-b,n)=1$. For $X_n$ the chromatic number, the clique number, the independence number, the diameter and the vertex connectivity are determined. We decide on the perfectness of $X_n$ and show that all nonzero eigenvalues of $X_n$ are integers dividing the value $\varphi(n)$ of the Euler function.

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.…”

“…For this reason, we record the following fact and omit the proof. Another standard result concerning unitary Cayley graphs states that the clique number and the chromatic number of G Z/(n) are both equal to the smallest prime factor of n [2]. Again, the proof generalized in a straightforward manner, so we omit the proof of the next fact.…”

“…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.…”

Enumerating cliques in direct product graphs

“…in turn, is also the chromatic number of G Z/nZ [23]. Klotz and Sander have also shown that the eigenvalues of G Z/nZ (that is, the eigenvalues of an adjacency matrix of G Z/nZ ) are integers that divide ϕ(n) (in fact, they are given by Ramanujan sums) [23]. Figure 1 depicts the graphs G Z/nZ for 2 ≤ n ≤ 10.…”

“…The clique number of G Z/nZ , defined to be the largest integer k such that G Z/nZ has a clique of order k, turns out to be the smallest prime factor of n. This, E-mail address: cdefant@princeton.edu. in turn, is also the chromatic number of G Z/nZ [23]. Klotz and Sander have also shown that the eigenvalues of G Z/nZ (that is, the eigenvalues of an adjacency matrix of G Z/nZ ) are integers that divide ϕ(n) (in fact, they are given by Ramanujan sums) [23].…”