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.

Two new graph characteristics, the total vertex irregularity strength and the total edge irregularity strength, are introduced. Estimations on these parameters are obtained. For some families of graphs the precise values of these parameters are proved.

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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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.

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