Symmetric property and reliability of locally twisted cubes

“…Thus, the only automorphism that fixes 0 is trivial and Det(LT Q n ) = 1, Dist(LT Q n ) = 2, and ρ(LT Q n ) = 1. By [11], when n = 3, Aut(LT Q 3 ) = D 16 , the automorphisms of the octagon. We conclude Det(LT Q 3 ) = Dist(LT Q 3 ) = 2 and ρ(LT Q 3 ) = 3.…”

Section: Locally Twisted Hypercubesmentioning

confidence: 99%

Symmetry parameters of various hypercube families

Boutin

Cockburn

Keough

et al. 2022

Art Discrete Appl. Math.

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“…For a graph Γ, a subgroup G of Aut(Γ) is semiregular on V (Γ) if evey element in G, except the identity, cannot fix a vertex of Γ, and regular if G is both transitive and semiregular on V (Γ). The graph Γ is a Cayley graph of a group G if there exists a regular subgroup of Aut(Γ) isomorphic to G (see [10,30]).…”

Section: Introductionmentioning

confidence: 99%

“…The class of m-Cayley graphs provides a useful tool to study non-vertex-transitive graphs, see [12] for example. It also has been used in the research of some networks, see [10,21] for example.…”

Section: Introductionmentioning

confidence: 99%

Symmetric property and edge-disjoint Hamiltonian cycles of the spined cube

Yang¹,

Zhang²,

Yan³

et al. 2022

Preprint

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The spined cube SQn is a variant of the hypercube Qn, introduced by Zhou et al. in [Information Processing Letters 111 (2011) 561-567] as an interconnection network for parallel computing. A graph Γ is an m-Cayley graph if its automorphism group Aut(Γ) has a semiregular subgroup acting on the vertex set with m orbits, and is a Caley graph if it is a 1-Cayley graph. It is well-known that Qn is a Cayley graph of an elementary abelian 2-group Z n 2 of order 2 n . In this paper, we prove that SQn is a 4-Cayley graph of Z n−2 2 when n ≥ 6, and is a ⌊n/2⌋-Cayley graph when n ≤ 5. This symmetric property shows that an n-dimensional spined cube with n ≥ 6 can be decomposed to eight vertex-disjoint (n − 3)-dimensional hypercubes, and as an application, it is proved that there exist two edge-disjoint Hamiltonian cycles in SQn when n ≥ 4. Moreover, we determine the vertex-transitivity of SQn, and prove that SQn is not vertex-transitive unless n ≤ 3.

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“…We let Aut(G) denote the group of all automorphisms of G. A graph G is vertex-transitive if for any pair of vertices u, v ∈ V (G), there is an automorphism σ ∈ Aut(G) such that σ(u) = v. One can also define edgetransitive, arc-transitive, and distance-transitive graphs; see [3] for precise definitions. Yes Yes Yes Yes [22] No [11] No [11] No No Table 1: Summary of the transitivity of various hypercube families (n ≥ 3).…”

Section: Introductionmentioning

confidence: 99%