We prove that the direct product of two coprime order elementary abelian groups of rank two, as well as the direct product of a cyclic group of prime order and a cyclic group of square-free order are ...DCI-groups. The latter is a generalization of Muzychuk's result on cyclic groups (J. Combin. Theory Ser. A, 1995).
In this paper we first determine all possible connected core-free 2-arc transitive Cayley graphs of the dicyclic group,
B
4
n
, and then show that this can be used to classify all connected 2-arc ...transitive Cayley graphs of this group in terms of regular cyclic covers, provided that we also know connected core-free 2-arc transitive Cayley graphs of the dihedral group.
Roughness in Fuzzy Cayley Graphs Shahzamanian, M.H.; Davvaz, B.
Қарағанды университетінің хабаршысы. Математика сериясы,
12/2023, Volume:
112, Issue:
4
Journal Article
Peer reviewed
Open access
Rough set theory is a worth noticing approach for inexact and uncertain system modelling. When rough set theory accompanies with fuzzy set theory, which both are a complementary generalization of set ...theory, they will be attended by potency in theoretical discussions. In this paper a definition for fuzzy Cayley subsets is put forward as well as fuzzy Cayley graphs of fuzzy subsets on groups inspired from the definition of Cayley graphs. We introduce rough approximation of a Cayley graph with respect to a fuzzy normal subgroup. We introduce the approximation rough fuzzy Cayley graphs and fuzzy rough fuzzy Cayley graphs. The last approximation is the mixture of the other approximations. Some theorems and properties are investigated and proved.
Let Fq be the finite field of order q and let Mm×n(Fq) be the additive (abelian) group consisting of all m×n matrices over Fq. Given an integer r with 0≤r≤min{m,n}, the Cayley graph G(m,n,r) is ...defined as the graph whose vertices are consisting of all the elements of Mm×n(Fq), and two vertices A,B∈Mm×n(Fq) are adjacent if the rank of A−B (denoted by rank(A−B)) is equal to r. In this paper, a recursion relation for the eigenvalues of G(m,n,r) is established; consequently, explicit formulas for all the eigenvalues of G(m,n,1) are exhibited immediately, which is a main result obtained previously in Delsarte (1975) 4.
Let G denote a dihedral group, where 1 is identity element and T⊆G∖{1}. We define T as minimal if T satisfies the condition 〈T〉=G, and there is an element s∈T satisfying 〈T∖{s,s−1}〉≠G. Within this ...manuscript, we achieve a complete characterization of the directed strongly regular Cayley graph Cay(G,T) of G, given the constraint that the subset T is minimal.
•We introduce the concept of “minimal directed strongly regular Cayley graphs,” simplifying and enhancing the investigation process.•This manuscript provides a comprehensive characterization of minimal directed strongly regular Cayley graphs over the dihedral group, marking the first contribution to this area. Indeed, the majority of known results regarding directed strongly regular graphs are constructive in nature.•The methodology presented in this paper holds applicability to numerous other non-abelian groups, including generalized dihedral groups, dicyclic groups, and beyond.
. 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 .
•We give a necessary and sufficient condition for the existence of efficient dominating sets of semi-Cayley graphs.•We give some results about the existence of efficient dominating sets of Cayley ...graphs over two non-abelian groups.•These results enrich the results of efficient dominating sets in Cayley graphs.
An independent perfect dominating set in a graph Γ with vertex set V(Γ) is a subset S of V(Γ) such that S is an independent set and every vertex in V(Γ)∖S is adjacent to exactly one vertex in S. In this paper, we first give a necessary and sufficient condition for the existence of independent perfect dominating sets in semi-Cayley graphs of finite groups. Further, we obtain a necessary and sufficient condition for Cayley graphs to have independent perfect dominating sets on two classes non-abelian groups.
