The dimension (or “minimal base size”) of a finite permutation group action is defined to be the smallest power of the action that contains a regular orbit. Although the concept has appeared before in various contexts, the intention of the current paper is to survey it from a slightly different viewpoint, with particular emphasis on its behaviour with respect to G-set constructions. Elementary ...

Let L X be a lamplighter graph, i.e., the graph-analogue of a wreath product of groups, and let P be the transition operator (matrix) of a random walk on that structure. We explain how methods developed by Saloff-Coste and the author can be applied for determining the p-norms and spectral radii of P , if one has an amenable (not necessarily discrete or unimodular) locally compact group of isome...

Various algebraic structures have recently appeared in a parallel way in the framework of Hilbert schemes of points on a surface and respectively in the framework of equivariant K-theory [N1, Gr, S2, W], but direct connections are yet to be clarified to explain such a coincidence. We provide several non-trivial steps toward establishing our main conjecture on the isomorphism between the Hilbert...

Let H be a finite group, Tn the symmetric semigroup of degree n, and let α, β be integers with 0 < α < β. We establish congruences modulo an arbitrary prime for the number sn(α, β,H) of solutions of the equationX α = X in the wreath product H oTn, where n = p, p+1, p+2, p+3. Our results generalise well-known congruences for the number of idempotent elements in Tn, to which they reduce for α = 1...

We show that the conjugacy problem in a wreath product A ≀B is uniformTC -Turing-reducible to the conjugacy problem in the factors A and B and the power problem in B. Moreover, if B is torsion free, the power problem for B can be replaced by the slightly weaker cyclic submonoid membership problem for B, which itself turns out to be uniform-TC-Turing-reducible to conjugacy problem in A ≀B if A i...

The question of when a given graph can be the underlying graph of a regular map has roots a hundred years old and is currently the object of several threads of research. This paper outlines this topic briefly and proves that a product of graphs which have regular embeddings also has such an embedding. We then present constructions for members of three families: (1) circulant graphs, (2) wreath ...

For any integer $k\geq 3$ , we define sunlet graph of order $2k$, denoted by $L_{2k}$, as the consisting a cycle length $k$ together with pendant vertices, each adjacent to exactly one vertex cycle. In this paper, give necessary and sufficient conditions for existence $L_{8}$-decomposition tensor product wreath complete graphs.

The base size of a permutation group, and the metric dimension of a graph, are both well-studied parameters which are closely related. We survey results on the relationship between the two, and with other, related parameters of groups, graphs, coherent configurations and association schemes. We also present some new results, including on the base sizes of wreath products in the product action, ...