Let ℳ be a regular map of genus g>1 and X be the underlying Riemann surface. A reflection of ℳ fixes some simple closed curves on X, which we call mirrors. Each mirror passes through at least two of the geometric points (vertices, face-centers and edge-centers) of ℳ. In this paper we study the surfaces which contain mirrors passing through just two geometric points, and show that only Wiman surfaces have this property.

A one-dimensional tiling is a bi-infinite string on a finite alphabet, and its tiling semigroup is an inverse semigroup whose elements are marked finite substrings of the tiling. We characterize the structure of these semigroups in the periodic case, in which the tiling is obtained by repetition of a fixed primitive word.

We prove that n-hypergraphs can be interpreted in e-free perfect PAC fields in particular in pseudofinite fields. We use methods of function field arithmetic, more precisely we construct generic polynomials with alternating groups as Galois groups over a function field.

Using the framework of overpartitions, we give a combinatorial interpretation and proof of the q-Bailey identity. We then deduce from this identity a couple of facts about overpartitions. We show that the method of proof of the q-Bailey identity also applies to the (first) q-Gauss identity.

We show that some q-series such as universal mock theta functions are linear sums of theta quotients and mock Jacobi forms of weight 1/2, which become holomorphic parts of real analytic modular forms when they are restricted to torsion points and multiplied by suitable powers of q. We also prove that certain linear sums of q-series are weakly holomorphic modular forms of weight 1/2 due to annihilation of mock Jacobi forms or completion by mock Jacobi forms. As an application, we obtain a relation between the rank and crank of a partition.

Using elementary means, we derive an explicit formula for a3(n), the number of 3-core partitions of n, in terms of the prime factorization of 3n+1. Based on this result, we are able to prove several infinite families of arithmetic results involving a3(n), one of which specializes to the recent result of Baruah and Berndt which states that, for all n≥0, a3(4n+1)=a3(n).

Qualified difference sets (QDS) composed of unions of cyclotomic classes are discussed. An exhaustive computer search for such QDS and modified QDS that also possess the zero residue has been conducted for all powers n=4,6,8 and 10. Two new families were discovered in the case n=8 and some new isolated systems were discovered for n=6 and n=10.

In this paper we give an analytic solution for graphs with n nodes and E = cn log n edges for which the probability of obtaining a given graph G is µn (G) = exp (- β ∑i=1ndi2), where di is the degree of node i. We describe how this model appears in the context of load balancing in communication networks, namely peer-to-peer overlays. We then analyse the degree distribution of such graphs and show that the degrees are concentrated around their mean value. Finally, we derive asymptotic results for the number of edges crossing a graph cut and use these results (i) to compute the graph expansion and conductance, and (ii) to analyse the graph resilience to random failures.

Let 𝒫 be a Poisson process of intensity 1 in a square Sn of area n. For a fixed integer k, join every point of 𝒫 to its k nearest neighbours, creating an undirected random geometric graph Gn,k. We prove that there exists a critical constant ccrit such that, for c < ccrit, Gn,⌊c log n⌋ is disconnected with probability tending to 1 as n → ∞ and, for c > ccrit, Gn,⌊c log n⌋ is connected with probability tending to 1 as n → ∞. This answers a question posed in Balister et al. (2005).

Let Γ be a finite connected undirected vertex transitive locally primitive graph of prime-power order. It is shown that either Γ is a normal Cayley graph of a 2-group, or Γ is a normal cover of a complete graph, a complete bipartite graph, or Σ×l, where Σ=Kpm with p prime or Σ is the Schläfli graph (of order 27). In particular, either Γ is a Cayley graph, or Γ is a normal cover of a complete bipartite graph.

The admissible representations of a real reductive group G are known by work of Langlands, Knapp, Zuckerman and Vogan. This paper describes an effective algorithm for computing the irreducible representations of G with regular integral infinitesimal character. The algorithm also describes structure theory of G, including the orbits of K(ℂ) (a complexified maximal compact subgroup) on the flag variety. This algorithm has been implemented on a computer by the second author, as part of the ‘Atlas of Lie Groups and Representations’ project.

Small-world graphs are examples of random graphs which mimic empirically observed features of social networks. We propose an intrinsic definition of small-world graphs, based on a probabilistic formulation of scaling properties of the graph, which does not rely on any particular construction. Our definition is shown to encompass existing models of small-world graphs, proposed by Watts (1999) and studied by Barbour and Reinert (2001), which are based on random perturbations of a regular lattice. We also propose alternative constructions of small-world graphs which are not based on lattices and study their scaling properties.

An S3-involution graph for a group G is a graph with vertex set a union of conjugacy classes of involutions of G such that two involutions are adjacent if they generate an S3-subgroup in a particular set of conjugacy classes. We investigate such graphs in general and also for the case where G=PSL(2,q).

We exhibit an interesting Cayley graph X of the elementary abelian group Z26 with the property that Aut(X) contains two regular subgroups, exactly one of which is normal. This demonstrates the existence of two subsets of Z26 that yield isomorphic Cayley graphs, even though the two subsets are not equivalent under the automorphism group of Z26.

The core of a graph Γ is the smallest graph Δ that is homomorphically equivalent to Γ (that is, there exist homomorphisms in both directions). The core of Γ is unique up to isomorphism and is an induced subgraph of Γ. We give a construction in some sense dual to the core. The hull of a graph Γ is a graph containing Γ as a spanning subgraph, admitting all the endomorphisms of Γ, and having as core a complete graph of the same order as the core of Γ. This construction is related to the notion of a synchronizing permutation group, which arises in semigroup theory; we provide some more insight by characterizing these permutation groups in terms of graphs. It is known that the core of a vertex-transitive graph is vertex-transitive. In some cases we can make stronger statements: for example, if Γ is a non-edge-transitive graph, we show that either the core of Γ is complete, or Γ is its own core. Rank-three graphs are non-edge-transitive. We examine some families of these to decide which of the two alternatives for the core actually holds. We will see that this question is very difficult, being equivalent in some cases to unsolved questions in finite geometry (for example, about spreads, ovoids and partitions into ovoids in polar spaces).

In this note we first prove that, for a positive integer n>1 with n≠p or p2 where p is a prime, there exists a transitive group of degree n without regular subgroups. Then we look at 2-closed transitive groups without regular subgroups, and pose two questions and a problem for further study.

A first kind Frobenius graph is a Cayley graph Cay(K,S) on the Frobenius kernel of a Frobenius group such that S=aH for some a∈K with 〈aH〉=K, where H is of even order or a is an involution. It is known that such graphs admit ‘perfect’ routing and gossiping schemes. A circulant graph is a Cayley graph on a cyclic group of order at least three. Since circulant graphs are widely used as models for interconnection networks, it is thus highly desirable to characterize those which are Frobenius of the first kind. In this paper we first give such a characterization for connected 4-valent circulant graphs, and then describe optimal routing and gossiping schemes for those which are first kind Frobenius graphs. Examples of such graphs include the 4-valent circulant graph with a given diameter and maximum possible order.

A systematic study of random Laguerre tessellations, weighted generalisations of the well-known Voronoi tessellations, is presented. We prove that every normal tessellation with convex cells in dimension three and higher is a Laguerre tessellation. Tessellations generated by stationary marked Poisson processes are then studied in detail. For these tessellations, we obtain integral formulae for geometric characteristics and densities of the typical k-faces. We present a formula for the linear contact distribution function and prove various limit results for convergence of Laguerre to Poisson-Voronoi tessellations. The obtained integral formulae are subsequently evaluated numerically for the planar case, demonstrating their applicability for practical purposes.

We count how many ‘different’ Morse functions exist on the 2-sphere. There are several ways of declaring that two Morse functions f and g are ‘indistinguishable’ but we concentrate only on two natural equivalence relations: homological (when the regular sublevel sets f and g have identical Betti numbers) and geometric (when f is obtained from g via global, orientation-preserving changes of coordinates on S2 and ℝ). The count of homological classes is reduced to a count of lattice paths confined to the first quadrant. The count of geometric classes is reduced to a count of certain labeled trees, which is encoded by certain elliptic integrals.