Permutability graphs of subgroups of some finite non-abelian groups

Abstract

In this paper, we study the structure of the permutability graphs of subgroups, and the permutability graphs of non-normal subgroups of the following groups: the dihedral groups Dn, the generalized quaternion groups Qn, the quasi-dihedral groups QD 2 n and the modular groups M p n . Further, we investigate the number of edges, degrees of the vertices, independence number, dominating number, clique number, chromatic number, weakly perfectness, Eulerianness, Hamiltonicity of these graphs.

Distance-regular Cayley graphs over dicyclic groups

“…Since m is a divisor of 4n, by elementary group theory, there exists a subgroup H (m) of order m in Dic n (cf. [19]). Then it is easy to see that K t×m ∼ = Cay(Dic n , Dic n \ H (m) ).…”

“…Also, it is easy to see that V (Γ + ) acts regularly on itself by left multiplication as a subgroup of Aut(Γ + ). Since every subgroup of Dic n is cyclic or dicyclic (cf [19]…”

The Intersection Graph of Subgroups of the Dihedral Group of Order 2pq

“…The intersection graph of is an undirected simple (without loops and multiple edges) graph whose vertex-set consists of all nontrivial proper subgroups of for which two distinct vertices H and K of are adjacent if ⋂ is a nontrivial subgroup of . This kind of graph has been studied by researchers; we refer the reader to see [2][3][4][5][6]. Let be any graph.…”