Rotational circulant graphs

Journal of Graph Theory

2015

Self Cite

A retract of a graph is an induced subgraph of such that there exists a homomorphism from to whose restriction to is the identity map. A graph is a core if it has no nontrivial retracts. In general, the minimal retracts of a graph are cores and are unique up to isomorphism; they are called the core of the graph. A graph is G-symmetric if G is a subgroup of the automorphism group of that is transitive on the vertex set and also transitive on the set of ordered pairs of adjacent vertices. If in addition the vertex set of admits a nontrivial partition that is preserved by G, then is an imprimitive G-symmetric graph. In this paper cores of imprimitive symmetric graphs of order a product of two distinct primes are studied. In many cases the core of is determined completely. In other cases it is proved that either is a core or its core is isomorphic to one of two graphs, and conditions on when each of these possibilities occurs is given.

“…Moreover,

G (p, r)

has valency r , and if

r < p - 1

then

Aut (G (p, r)) ≅ {double-struckZ}_{p} ⋊ H (p, r)

(

\leq AGL (1, p)

) is a Frobenius group in its action on the vertex set

Z_{p}

G (p, r)

, while

G (p, p - 1) = K_{p}

. (The fact that

Aut (G (p, r))

is a Frobenius group on

Z_{p}

was also observed in [, corollary 2.11] in a different setting.) Definition Let A and

A^{'}

be two disjoint copies of

Z_{q}

, and for each

i \in {double-struckZ}_{q}

, denote the corresponding elements of A and

A^{'}

by i and

i^{'}

, respectively.…”

Section: Cores Of Imprimitive Symmetric Circulant Graphs Of Order Pqmentioning

confidence: 69%

Cores of Imprimitive Symmetric Graphs of Order a Product of Two Distinct Primes

Rotheram

Journal of Graph Theory

2015

Self Cite

“…In this section we construct RMDs from cyclic groups by using Theorem 3.1 and recent results [28][29][30]36] on first-kind Frobenius circulant graphs. As usual we denote the additive group of integers modulo n by Z n and the multiplicative group of units of the ring Z n by Z * n = {[u] : 1 ≤ u ≤ n − 1, gcd(n, u) = 1}.…”

Section: Constructing Resolvable Mendelsohn Designs From Cyclic Groupsmentioning

confidence: 99%

“…To illustrate this method, we will use Theorem 3.1 and a known result [12] on Ferrero pairs to prove the existence and give an explicit construction of a (v, k, 1)-RPMD, for any integer with prime factorisation v = p e 1 1 • • • p e t t and any prime k dividing every p e i i − 1 (see Theorem 3.8). We will also use Theorem 3.1 and a recent result from network design [30] [22] and motivated by the study of Latin squares, there is a long history of studying complete mappings. Hall and Paige [19] proved that a finite group with a nontrivial, cyclic Sylow 2-subgroup does not admit complete mappings.…”

Section: Introductionmentioning

confidence: 99%

Resolvable Mendelsohn Designs and Finite Frobenius Groups

Hsu

2018

Bull. Aust. Math. Soc.

Self Cite

We prove the existence and give constructions of a (p(k) − 1)-fold perfect resolvable (v, k, 1)-Mendelsohn design for any integers v > k ≥ 2 with v ≡ 1 mod k such that there exists a finite Frobenius group whose kernel K has order v and whose complement contains an element φ of order k, where p(k) is the least prime factor of k. Such a design admits K ⋊ φ as a group of automorphisms and is perfect when k is a prime. As an application we prove that for any integer v = p e1 1 . . . p et t ≥ 3 in prime factorization, and any prime k dividing p ei i − 1 for 1 ≤ i ≤ t, there exists a resolvable perfect (v, k, 1)-Mendelsohn design that admits a Frobenius group as a group of automorphisms. We also prove that, if k is even and divides p i − 1 for 1 ≤ i ≤ t, then there are at least ϕ(k) t resolvable (v, k, 1)-Mendelsohn designs that admit a Frobenius group as a group of automorphisms, where ϕ is Euler's totient function.

“…This has been achieved in [33] and [36] in the case of valency 4 and 6, respectively. (See also [34,35,41] for related results.) In this paper we classify first kind Frobenius circulants of valency 2p for any odd prime p (Theorem 5.3), and prove that all of them are p th cyclotomic graphs (Theorem 5.5).…”

Section: Introductionmentioning

confidence: 99%

“…Conversely, if n ≡ 1 mod 2p and a is a positive integer such that a p + 1 ≡ 0 mod n and a i ± 1, 1 ≤ i ≤ p − 1 are coprime to n, then H = [a] ≤ Z * n is semiregular on Z n \ {[0]} with order |H| = 2p. Therefore, Z n H is a Frobenius group and Cay(Z n , [a] ) is a first kind [35,Corollary 2.9]). Note that Cay(Z n , [a] ) is a rotational Cayley graph.…”

mentioning

confidence: 99%

Cyclotomic graphs and perfect codes

Journal of Pure and Applied Algebra

2019

Self Cite

We study two families of cyclotomic graphs and perfect codes in them. They are Cayley graphs on the additive group ofis an mth primitive root of unity, A a nonzero ideal of Z[ζ m ], and φ Euler's totient function. We call them the mth cyclotomic graph and the second kind mth cyclotomic graph, and denote them by G m (A) and G * m (A), respectively. We give a necessary and sufficient condition for D/A to be a perfect t-code in G * m (A) and a necessary condition for D/A to be such a code in G m (A), where t ≥ 1 is an integer and D an ideal of Z[ζ m ] containing A. In the case when m = 3, 4, G m ((α)) is known as an Eisenstein-Jacobi and Gaussian networks, respectively, and we obtain necessary conditions for (β)/(α) to be a perfect t-code in G m ((α)), where 0 = α, β ∈ Z[ζ m ] with β dividing α. In the literature such conditions are known to be sufficient when m = 4 and m = 3 under an additional condition. We give a classification of all first kind Frobenius circulants of valency 2p and prove that they are all pth cyclotomic graphs, where p is an odd prime. Such graphs belong to a large family of Cayley graphs that are efficient for routing and gossiping.