A canonical double cover B(X) of a graph X is the direct product of X and the complete graph K2 on two vertices. In order to answer the question when a canonical double cover of a given graph is a ...Cayley graph, in 1992 Marušič et al. introduced the concept of generalized Cayley graphs. In this paper this concept is generalized to a wider class of graphs, the so-called extended generalized Cayley graphs. It is proved that the canonical double cover of a connected non-bipartite graph X is a Cayley graph if and only if X is an extended generalized Cayley graph. This corrects an incorrectly stated claim in Discrete Math. 102 (1992), 279–285.
GCI-property of some groups Liao, Qianfen; Liu, Weijun
Applied mathematics and computation,
02/2023, Volume:
438
Journal Article
Peer reviewed
In this paper, firstly, we determine the local 2-GCI-property and 2−GCI-property of the cyclic group. Then, for the dihedral group D2n, we prove that it has local GCI-property if and only if n is an ...odd prime or 9. Further, the dihedral group D2n cannot have GCI-property. Moreover, we discuss the GCI-property of the elementary abelian group, the dicyclic group and the semi-dihedral group.
The isomorphism problem is a fundamental problem for algebraic and combinatorial structures, particularly in relation to Cayley graphs. Let Xi=GC(G,Si,αi),(i=1,2) be generalized Cayley graphs. If ...whenever X1≅X2, it implies that α2=α1γ and S2=g−1S1γgα2 for some g∈G and γ∈Aut(G), then G is a strongly generalized Cayley isomorphism (GCI)-group. In this study, we defined (strongly, restricted) m-GCI-groups. These definitions are similar to those of m-CI-groups for Cayley graphs. Our main results demonstrate that a finite non-abelian simple group G is a restricted 2-GCI-group if and only if G is one of A5, L2(8), M11, Sz(8), or M23, and G is a 2-GCI-group if and only if G is A5 or L2(8).
. Suppose that is the Cayley graph whose vertices are all elements of and two vertices and are adjacent if and only if . In this paper,we introduce the generalized Cayley graph denoted by ... which is a graph with a vertex set consisting of all column matrices in which all components are in and two vertices and are adjacent if and only if , where is a column matrix that each entry is the inverse of the similar entry of and is matrix with all entries in , is the transpose of and and m . We aim to provide some basic properties of the new graph and determine the structure of when is a complete graph for every , and n, m .
Suppose that is a finite group and is a non-empty subset of such that and . Suppose that is the Cayley graph whose vertices are all elements of and two vertices and are adjacent if and only ...if . In this paper, we introduce the generalized Cayley graph denoted by that is a graph with vertex set consists of all column matrices which all components are in and two vertices and are adjacent if and only if , where is a column matrix that each entry is the inverse of similar entry of and is matrix with all entries in , is the transpose of and . In this paper, we clarify some basic properties of the new graph and assign the structure of when is complete graph , complete bipartite graph and complete 3-partite graph for every .
Generalized Cayley graphs of semigroups were first defined by Zhu (Semigroup Forum 84:131–143,
2012
) and further studied by Zhu (Semigroup Forum 84:144–156,
2012
) as well as by Wang (Semigroup ...Forum 86:221–223,
2013
). In the present paper, we study the vertex-transitivity of generalized Cayley graphs of a semigroup so that two main theorems of Kelarev and Praeger (Eur J Comb 24:59–72,
2003
) for the classical and well known Cayley graphs of semigroups are extended to those for generalized Cayley graphs of a semigroup.
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