The degree/diameter problem is to determine the largest graphs or digraphs of given maximum degree and given diameter.General upper bounds - called Moore bounds - for the order of such graphs and digraphs are attainable only for certain special graphs and digraphs. Finding better (tighter) upper bounds for the maximum possible number of vertices, given the other two parameters, and thus attacking the degree/diameter problem 'from above', remains a largely unexplored area. Constructions producing large graphs and digraphs of given degree and diameter represent a way of attacking the degree/diameter problem 'from below'.This survey aims to give an overview of the current state-of-the-art of the degree/diameter problem. We focus mainly on the above two streams of research. However, we could not resist mentioning also results on various related problems. These include considering Moore-like bounds for special types of graphs and digraphs, such as vertex-transitive, Cayley, planar, bipartite, and many others, on the one hand, and related properties such as connectivity, regularity, and surface embeddability, on the other hand.

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Let vt(d, 2) be the largest order of a vertex-transitive graph of degree d and diameter 2. It is known that vt(d, 2)=d 2 +1 for d=1, 2, 3, and 7; for the remaining values of d we have vt(d, 2) d 2 &1. The only known general lower bound on vt(d, 2), valid for all d, seems to be vt(d, 2) w(d+2)Â2x W(d+2)Â2X. Using voltage graphs, we construct a family of vertex-transitive non-Cayley graphs which shows that vt(d, 2) (8Â9)(d+ 2 for all d of the form d=(3q&1)Â2, where q is a prime power congruent with 1 (mod 4). The construction generalizes to all prime powers and yields large highly symmetric graphs for other degrees as well. In particular, for d=7 we obtain as a special case the Hoffman Singleton graph, and for d=11 and d=13 we have new largest graphs of diameter 2, and degree d on 98 and 162 vertices, respectively.

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An enumeration result for orientably regular hypermaps of a given type with automorphism groups isomorphic to PSL(2, q) or PGL(2, q) can be extracted from a 1969 paper by Sah. We extend the investigation to orientable reflexible hypermaps and to nonorientable regular hypermaps, providing many more details about the associated computations and explicit generating sets for the associated groups.2000 Mathematics subject classification: primary 57M15; secondary 05C25, 20F05. Keywords and phrases: hypermap, regular hypermap, triangle group, projective linear group. IntroductionA regular hypermap H is a pair (r, s) of permutations generating a regular permutation group on a finite set, and provides a generalization of the geometric notion of a regular map on a surface, by allowing edges to be replaced by 'hyperedges'. The cycles of r, s and r s correspond to the hypervertices, hyperedges and hyperfaces of H, which determine the embedding of the underlying (and connected) hypergraph into the surface, and their orders give the type of H, say {k, l, m}. The group G generated by r and s induces a group of automorphisms of this hypergraph, preserving the embedding, and acting transitively on the flags (incident hypervertex-hyperedge pairs) of H. When one of the parameters k, l, m is 2, the hypergraph is a graph, and the hypermap is a regular map. The theory of such objects is well developed, and has been thoroughly explained in [10,11]. Without going into too much detail, we need to make a few basic observations. First, the group G has a presentation of the formand (so) is a finite quotient of the ordinary (k, l, m) triangle group. For simplicity, we will say that such a group G has type (k, l, m), provided that k, l, m are the true orders of the corresponding elements r, s, t. There is a bijective correspondence between isomorphism classes of regular hypermaps of a given type {k, l, m} and torsionfree normal subgroups of the ordinary (k, l, m) triangle group (k, l, m), and the number of those with a given group G as 'rotational symmetry group' (or quotient of (k, l, m)) is equal to the number of ways of generating G by a (k, l, m)-triple (r, s, t) up to equivalence under Aut(G). For further details about representing hypermaps in the form of cellular decomposition of closed two-dimensional surfaces and visualizing the rotational symmetries, and also their association with Riemann surfaces and algebraic number fields (through Grothendieck's theory of dessins d'enfants), we refer the reader to [4,10,11].A regular hypermap may admit a symmetry that induces a reversal of some local orientation of the supporting surface. At the group theory level, this is equivalent to the existence of an automorphism ϑ of a (k, l, m)-group G presented as above, such that ϑ inverts two of the three generators. Such regular hypermaps are called reflexible. If ϑ is actually given by conjugation of some element of order two in G, then the corresponding (k, l, m)-generating triple for G gives rise to two distinct reflexible hypermaps: one ...

Abstract. This paper uses combinatorial group theory to help answer some long-standing questions about the genera of orientable surfaces that carry particular kinds of regular maps. By classifying all orientably-regular maps whose automorphism group has order coprime to g −1, where g is the genus, all orientably-regular maps of genus p+1 for p prime are determined. As a consequence, it is shown that orientable surfaces of infinitely many genera carry no regular map that is chiral (irreflexible), and that orientable surfaces of infinitely many genera carry no reflexible regular map with simple underlying graph. Another consequence is a simpler proof of the Breda-Nedela-Širáň classification of non-orientable regular maps of Euler characteristic −p where p is prime.

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