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Register a new userA classification of the m-graphical regular representation of finite groups
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In this paper we extend the classical notion of digraphical and graphical regular representation of a group and we classify, by means of an explicit description, the finite groups satisfying this generalization. A graph or digraph is called regular if each vertex has the same valency, or, the same out-valency and the same in-valency, respectively. An m-(di)graphical regular representation (respectively, m-GRR and m-DRR, for short) of a group G is a regular (di)graph whose automorphism group is isomorphic to G and acts semiregularly on the vertex set with m orbits. When m=1, this definition agrees with the classical notion of GRR and DRR. Finite groups admitting a 1-DRR were classified by Babai in 1980, and the analogue classification of finite groups admitting a 1-GRR was completed by Godsil in 1981. Pivoting on these two results in this paper we classify finite groups admitting an m-GRR or an m-DRR, for arbitrary positive integers m. For instance, we prove that every non-identity finite group admits an m-GRR, for every m>4.
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We say that a finite group G is DRR-detecting if, for every subset S of G, either the Cayley digraph Cay(G,S) is a digraphical regular representation (that is, its automorphism group acts regularly on its vertex set) or there is a nontrivial group automorphism phi of G such that phi(S) = S. We show that every nilpotent DRR-detecting group is a p-group, but that the wreath product of two cyclic groups of order p is not DRR-detecting, for every odd prime p. We also show that if G and H are nontrivial groups that admit a digraphical regular representation and either gcd(|G|,|H|) = 1, or H is not DRR-detecting, then the direct product G x H is not DRR-detecting. Some of these results also have analogues for graphical regular representations.
Let $R$ be a commutative ring with identity. We define a graph $Gamma_{aut}(R)$ on $ R$, with vertices elements of $R$, such that any two distinct vertices $x, y$ are adjacent if and only if there exists $sigma in aut$ such that $sigma(x)=y$. The idea is to apply graph theory to study orbit spaces of rings under automorphisms. In this article, we define the notion of a ring of type $n$ for $ngeq 0$ and characterize all rings of type zero. We also characterize local rings $(R,M) $ in which either the subset of units ($ eq 1 $) is connected or the subset $M- {0}$ is connected in $Gamma_{aut}(R)$.
In this paper, we classify regular polytopes with automorphism groups of order $2^n$ and Schlafli types ${4, 2^{n-3}}, {4, 2^{n-4}}$ and ${4, 2^{n-5}}$ for $n geq 10$, therefore giving a partial answer to a problem proposed by Schulte and Weiss in [Problems on polytopes, their groups, and realizations, Periodica Math. Hungarica 53(2006) 231-255].
Let $G$ be a finite group. We will say that $M$ and $S$ form a textsl{complete splitting} (textsl{splitting}) of $G$ if every element (nonzero element) $g$ of $G$ has a unique representation of the form $g=ms$ with $min M$ and $sin S$, and $0$ has a such representation (while $0$ has no such representation). In this paper, we determine the structures of complete splittings of finite abelian groups. In particular, for complete splittings of cyclic groups our description is more specific. Furthermore, we show some results for existence and nonexistence of complete splittings of cyclic groups and find a relationship between complete splittings and splittings for finite groups.
A set $X$ in the Euclidean space $mathbb{R}^d$ is called an $m$-distance set if the set of Euclidean distances between two distinct points in $X$ has size $m$. An $m$-distance set $X$ in $mathbb{R}^d$ is said to be maximal if there does not exist a vector $x$ in $mathbb{R}^d$ such that the union of $X$ and ${x}$ still has only $m$ distances. Bannai--Sato--Shigezumi (2012) investigated the maximal $m$-distance sets which contain the Euclidean representation of the Johnson graph $J(n,m)$. In this paper, we consider the same problem for the Hamming graph $H(n,m)$. The Euclidean representation of $H(n,m)$ is an $m$-distance set in $mathbb{R}^{m(n-1)}$. We prove that the maximum $n$ is $m^2 + m - 1$ such that the representation of $H(n,m)$ is not maximal as an $m$-distance set. Moreover we classify the largest $m$-distance sets which contain the representation of $H(n,m)$ for $mleq 4$ and any $n$. We also classify the maximal $2$-distance sets in $mathbb{R}^{2n-1}$ which contain the representation of $H(n,2)$ for any $n$.
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