Related papers: Transversals, plexes, and multiplexes in iterated …

Let X be a finite set of complex numbers and let A be a normal operator with spectrum X that acts on a separable Hilbert space H. Relative to a fixed orthonormal basis e_1,e_2, ... for H, A gives rise to a matrix whose diagonal is a…

A Cayley digraph Cay(G,S) of a finite group $G$ with respect to a subset $S$ of $G$ is said to be a CI-digraph if for every Cayley digraph Cay(G,T) isomorphic to Cay(G,S), there exists an automorphism $\sigma$ of $G$ such that $S^\sigma=T$.…

Let G be a Chevalley group scheme of rank l. We show that the following holds for some absolute constant d>0 and two functions p_0=p_0(l) and C=C(l,p). Let p>p_0 be a prime number and let G_n:=G(\Z/p^n\Z) be the family of finite groups for…

An interesting fact is that most of the known connected $2$-arc-transitive nonnormal Cayley graphs of small valency on finite simple groups are $(\mathrm{A}_{n+1},2)$-arc-transitive Cayley graphs on $\mathrm{A}_n$. This motivates the study…

Let $G$ be a linear algebraic group over an infinite field $k$. Loosely speaking, a $G$-torsor over $k$-variety is said to be versal if it specializes to every $G$-torsor over any $k$-field. The existence of versal torsors is well-known. We…

For a graph $G$ and partition $\mathcal{U}$ of its vertex set, an independent transversal of $(G, \mathcal{U})$ is an independent set of $G$ that contains one vertex from each block of $\mathcal{U}$. Buys, Kang, and Ozeki studied when a…

Partial cubes are the graphs which can be embedded into hypercubes. The {\em cube polynomial} of a graph $G$ is a counting polynomial of induced hypercubes of $G$, which is defined as $C(G,x):=\sum_{i\geqslant 0}\alpha_i(G)x^i$, where…

We show that if G is a nontrivial, finite group of odd order, whose commutator subgroup [G,G] is cyclic of order p^m q^n, where p and q are prime, then every connected Cayley graph on G has a hamiltonian cycle.

In this paper we enumerate $k$-noncrossing tangled-diagrams. A tangled-diagram is a labeled graph whose vertices are $1,...,n$ have degree $\le 2$, and are arranged in increasing order in a horizontal line. Its arcs are drawn in the upper…

A graph $\Gamma$ is called $(G, s)$-arc-transitive if $G \le \mathrm{Aut}(\Gamma)$ is transitive on the set of vertices of $\Gamma$ and the set of $s$-arcs of $\Gamma$, where for an integer $s \ge 1$ an $s$-arc of $\Gamma$ is a sequence of…

A Cayley graph $\Cay(G,S)$ is said to be inner-automorphic if $S$ is a union of conjugacy classes of a group $G$, and arc-transitive if its full automorphism group acts transitively on the set of arcs. In this paper, we characterize four…

We show that every finitely generated group G with an element of order at least $(5rank(G))^{12}$ admits a locally finite directed Cayley graph with automorphism group equal to G. If moreover G is not generalized dihedral, then the above…

Let $p$ be an odd prime, and $D_{2p}=\langle a,b\mid a^p=b^2=1,bab=a^{-1}\rangle$ the dihedral group of order $2p$. In this paper, we completely classify the cubic Cayley graphs on $D_{2p}$ up to isomorphism by means of spectral method. By…

A finite group $G$ is a called a DCI-group if any two isomorphic Cayley digraphs of $G$ are also isomorphic via an automorphism of $G$. If $G$ is a non-abelian generalised dihedral DCI-group, then Dobson, Muzychuk, and Spiga proved that $G$…

An orientation of a graph is semi-transitive if it is acyclic, and for any directed path $v_0\rightarrow v_1\rightarrow \cdots\rightarrow v_k$ either there is no edge between $v_0$ and $v_k$, or $v_i\rightarrow v_j$ is an edge for all…

A two-dimensional simplicial complex is called $d$-{\em regular} if every edge of it is contained in exactly $d$ distinct triangles. It is called $\epsilon$-expanding if its up-down two-dimensional random walk has a normalized maximal…

We show that the braid group associated to the complex reflection group $G(d,d,n)$ is an index $d$ subgroup of the braid group of the orbifold quotient of the complex numbers by a cyclic group of order $d$. We also give a compatible…

Given a graph of groups $\mathcal{G} = (\Gamma, \{G_v\}, \{G_e\})$ with certain conditions on vertex groups and $G$ acts acylindrically on its Bass-Serre tree $T$. Let $H$ be a finitely generated subgroup of $G$. We prove the following…

The Kneser graph $K(n, k)$ has as vertices all $k$-element subsets of $[n]=\{1,2,...,n \}$ and an edge between any two vertices that are disjoint. If $n=2k+1$, then $K(n, k)$ is called an odd graph. Let $ n >4$ and $1< k < \frac{n}{2} $. In…

We prove that quadratical quasigroups form a variety Q of right and left simple groupoids. New examples of quadratical quasigroups of orders 25 and 29 are given. The fine structure of quadratical quasigroups and inter-relationships between…