A digraph $\Gamma$ is called $n$-Cayley digraph over a group $G$, if there exists a semiregular subgroup $R_G$ of Aut$(\Gamma)$ isomorphic to $G$ with $n$ orbits. In this paper, we represent the adjacency matrix of $\Gamma$ as a diagonal block matrix in terms of irreducible representations of $G$ and determine its characteristic polynomial. As corollaries of this result we find: the spectrum of semi-Cayley graphs over abelian groups, a relation between the characteristic polynomial of an $n$-Cayley graph and its complement, and the spectrum of Calye graphs over groups with cyclic subgroups. Finally we determine the eigenspace of $n$-Cayley digraphs and their main eigenvalues.
Let [Formula: see text] be any group and [Formula: see text] be a subgroup of [Formula: see text] for some set [Formula: see text]. The [Formula: see text]-closure of [Formula: see text] on [Formula: see text], denoted by [Formula: see text], is by definition, [Formula: see text] The group [Formula: see text] is called [Formula: see text]-closed on [Formula: see text] if [Formula: see text]. We say that a group [Formula: see text] is a totally[Formula: see text]-closed group if [Formula: see text] for any set [Formula: see text] such that [Formula: see text]. Here we show that the center of any finite totally 2-closed group is cyclic and a finite nilpotent group is totally 2-closed if and only if it is cyclic or a direct product of a generalized quaternion group with a cyclic group of odd order.
The full symmetry groups of carbon nanotori are investigated. It is shown that that the symmetry group of a chiral (n1, n2) nanotorus is isomorphic to D(2mq/n), where m and q are the number of lattice points on the torus circumference vector and the number of graphene hexagons in the nanotorus unit cell, respectively, and n = gcd(n1, n2). It is also shown that the symmetry group of zigzag and armchair (achiral) nanotori is D4m x Z2, where D2k and Zk are the dihedral group of order 2k and the cyclic group of order k, respectively. The irreducible representations and characters of these groups are discussed.
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