arXiv subjects
Publications and source records attributed to J. Siemons.
Let $Γ$ be a simple undirected graph on a finite vertex set and let $A$ be its adjacency matrix. Then $Γ$ is {\it singular} if $A$ is singular. The problem of characterising singular graphs is easy to state but very difficult to resolve in any generality. In this paper we investigate the singularity of graphs for which the dihedral group acts transitively on vertices as a group of automorphisms.
For a set $Ω$ an unordered relation on $Ω$ is a family R of subsets of $Ω.$ If R is such a relation we let G(R) be the group of all permutations on $Ω$ that preserves R, that is g belongs to G(R) if and only if x in R implies x^{g}\in R. We are interested in permutation groups which can be represented as G=G(R) for a suitable unordered relation R on $Ω.$ When this is the case, we say that G is defined by the relation R, or that G is a relation group. We prove that a primitive permutation group different from the Alternating Group and of degree bigger or equal to 11 is a relation groups. The same is true for many classes of finite imprimitive groups, and we give general conditions on the size of blocks of imprmitivity, and the groups induced on such blocks, which guarantee that the group is defined by a relation. This property is closely connected to the orbit closure of permutation groups. Since relation groups are orbit closed the results here imply that many classes of imprimitive permutation groups are orbit closed.