We present an enumeration of orientably-regular maps with automorphism group isomorphic to the twisted linear fractional group M (q 2 ) for any odd prime power q.
The degree-diameter problem seeks to find the maximum possible order of a graph with a given (maximum) degree and diameter. It is known that graphs attaining the maximum possible value (the Moore bound ) are extremely rare, but much activity is focussed on finding new examples of graphs or families of graph with orders approaching the bound as closely as possible.There has been recent interest in this problem as it applies to mixed graphs, in which we allow some of the edges to be undirected and some directed. A 2008 paper of Nguyen and Miller derived an upper bound on the possible number of vertices of such graphs. We show that for diameters larger than three, this bound can be reduced and we present a corrected Moore bound for mixed graphs, valid for all diameters and for all combinations of undirected and directed degrees.
The undirected degree/diameter and degree/girth problems and their directed analogues have been studied for many decades in the search for efficient network topologies. Recently such questions have received much attention in the setting of mixed graphs, i.e. networks that admit both undirected edges and directed arcs. The degree/diameter problem for mixed graphs asks for the largest possible order of a network with diameter k, maximum undirected degree ≤ r and maximum directed out-degree ≤ z. It is also of interest to find smallest possible k-geodetic mixed graphs with minimum undirected degree ≥ r and minimum directed out-degree ≥ z. A simple counting argument reveals the existence of a natural bound, the Moore bound, on the order of such graphs; a graph that meets this limit is a mixed Moore graph. Mixed Moore graphs can exist only for k = 2 and even in this case it is known that they are extremely rare. It is therefore of interest to search for graphs with order one away from the Moore bound. Such graphs must be out-regular; a much more difficult question is whether they must be totally regular. For k = 2, we answer this question in the affirmative, thereby resolving an open problem stated in a recent paper of López and Miret. We also present partial results for larger k. We finally put these results to practical use by proving the uniqueness of a 2-geodetic mixed graph with order exceeding the Moore bound by one.
We consider the problem of embedding a symmetric configuration with block size 3 in an orientable surface in such a way that the blocks of the configuration form triangular faces and there is only one extra large face. We develop a sufficient condition for such an embedding to exist given any orientation of the configuration, and show that this condition is satisfied for all configurations on up to 19 points. We also show that there exists a configuration on 21 points which is not embeddable in any orientation. As a by-product, we give a revised table of numbers of configurations, correcting the published figure for 19 points. We give a number of open questions about embeddability of configurations on larger numbers of points.
a b s t r a c tWe consider the degree-diameter problem for Cayley graphs of dihedral groups. We find upper and lower bounds on the maximum number of vertices of such a graph with diameter 2 and degree d. We completely determine the asymptotic behaviour of this class of graphs by showing that both limits are asymptotically d 2 /2.
The degree-diameter problem seeks to find the largest possible number of vertices in a graph having given diameter and given maximum degree. There has been much recent interest in the problem for mixed graphs, where we allow both undirected edges and directed arcs in the graph. For a diameter 2 graph with maximum undirected degree r and directed out-degree z, a straightforward counting argument yields an upper bound M (z, r, 2) = (z + r) 2 + z + 1 for the order of the graph. Apart from the case r = 1, the only three known examples of mixed graphs attaining this bound are Cayley graphs, and there are an infinite number of feasible pairs (r, z) where the existence of mixed Moore graphs with these parameters is unknown. We use a combination of elementary group-theoretical arguments and computational techniques to rule out the existence of further examples of mixed Cayley graphs attaining the Moore bound for all orders up to 485. PreliminariesThe degree-diameter problem has its roots in the efficient design of interconnection networks. We try to find the maximum possible number of vertices in a graph where we constrain both the largest degree of any vertex and the diameter of the graph. For more information on the history and development of the degree-diameter problem, see the survey [11]. The degree-diameter problem is typically studied in both the undirected and directed cases. Recently, there has been much interest in the problem as it is applied to mixed graphs, where we allow both undirected edges and directed arcs in the graph.In the undirected case, an upper bound for the largest possible order of a graph of maximum degree d > 2 and diameter k > 1 is the Moore bound [11]:Mathematics subject classification: 05C25, 05C35
The degree/diameter problem for mixed graphs asks for the largest possible order of a mixed graph with given diameter and degree parameters. Similarly the degree/geodecity problem concerns the smallest order of a k-geodetic mixed graph with given minimum undirected and directed degrees; this is a generalisation of the classical degree/girth problem. In this paper we present new bounds on the order of mixed graphs with given diameter or geodetic girth and exhibit new examples of directed and mixed geodetic cages. In particular, we show that any k-geodetic mixed graph with excess one must have geodetic girth two and be totally regular, thereby proving an earlier conjecture of the authors.
Regular maps on linear fractional groups PSL(2, q) and PGL(2, q) have been studied for many years and the theory is well-developed, including generating sets for the associated groups. This paper studies the properties of self-duality, self-Petrie-duality and Möbius regularity in this context, providing necessary and sufficient conditions for each case. We also address the special case for regular maps of type (5, 5). The final section includes an enumeration of the PSL(2, q) maps for q ≤ 81 and a list of all the PSL(2, q) maps which have any of these special properties for q ≤ 49.
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