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.
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.
•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.
We define the signed Cayley graph on Cayley graph Xn denoted by Sn, and study several properties such as balancing, clusterability and sign-compatibility of the signed Cayley graph Sn. Apart from it ...we also study the characterization of the canonical consistency of Sn, for some n.
A graph is half-arc-transitive if its full automorphism group acts transitively on vertices and edges, but not on arcs. Let p be a prime. It is known that there exists no tetravalent ...half-arc-transitive graph of order p or p2. All the tetravalent half-arc-transitive graphs of order p3 or p4 have been classified in two previous papers 9,23. As a continuation, in this paper, a classification is given of all tetravalent half-arc-transitive graphs of order p5.
An (r,z,k)-mixed graph G has every vertex with undirected degree r, directed in- and out-degree z, and diameter k. In this paper, we study the case r=z=1, proposing some new constructions of ...(1,1,k)-mixed graphs with a large number of vertices N. Our study is based on computer techniques for small values of k and the use of graphs on alphabets for general k. In the former case, the constructions are either Cayley or lift graphs. In the latter case, some infinite families of (1,1,k)-mixed graphs are proposed with diameter of the order of 2log2N.
