The following problem is considered: if $H$ is a semiregular abelian subgroup of a transitive permutation group $G$ acting on a finite set $X$, find conditions for (non) existence of $G$-invariant partitions of $X$. Conditions presented in this paper are derived by studying spectral properties of associated $G$-invariant digraphs. As an essential tool, irreducible complex characters of $H$ are used. Questions of this kind arise naturally when classifying combinatorial objects which enjoy a certain degree of symmetry. As an illustration, a new and short proof of an old result of Frucht, Graver and Watkins classifying edge-transitive generalized Petersen graphs, is given.

COBISS.SI-ID: 1536772036

Let $G$ denote a finite abelian group with identity 1 and let $S$ denote an inverse-closed subset of $G \setminus \{1\}$, which generates $G$ and for which there exists $s \in S$, such that $\langle S \setminus \{s,s^{-1}\} \rangle \ne G$. In this paper we obtain the complete classification of distance-regular Cayley graphs $\text{Cay}(G;S)$ for such pairs of $G$ and $S$.

COBISS.SI-ID: 1536382660

Let Γ denote a bipartite distance-regular graph with vertex set $X$ and diameter $D \ge 3$. Fix $x \in X$ and let $L$ (resp. $R$) denote the corresponding lowering (resp. raising) matrix. We show that each $Q$-polynomial structure for Γ yields a certain linear dependency among $RL^2$, $LRL$, $L^2R$, $L$. Define a partial order $\le$ on $X$ as follows. For $y,z \in X$ let $y \le z$ whenever $\partial(x,y)+\partial(y,z)=\partial(x,z)$, where $\partial$ denotes path-length distance. We determine whether the above linear dependency gives this poset a uniform or strongly uniform structure. We show that except for one special case a uniform structure is attained, and except for three special cases a strongly uniform structure is attained.

COBISS.SI-ID: 1024466772

Let Γ be a connected $G$-arc-transitive graph, let $uv$ be an arc of Γ and let $L$ be the permutation group induced by the action of the vertex-stabiliser $G_v$ on the neighbourhood $Γ(v)$. We study the problem of bounding $|G_{uv}|$ in terms of $L$ and the order of Γ.

COBISS.SI-ID: 16981849

A graph Γ is said to be G-arc-regular if a subgroup G≤Aut(Γ) acts regularly on the arcs of Γ. In this paper connected G-arc-regular graphs are classified in the case when G contains a regular dihedral subgroup D_2n of order 2n whose cyclic subgroup C_n ≤D_2n of index 2 is core-free in G. As an application, all regular Cayley maps over dihedral groups D_2n, n odd, are classified.

COBISS.SI-ID: 1024473940