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Introduction
The degree/diameter problem for bipartite graphs can be stated as follows:
Given natural numbers d and k, find the largest possible number N^{b}(d,k) of vertices in a bipartite graph of maximum degree d and diameter k.
An upper bound for N^{b}(d,k) is given by the socalled bipartite Moore bound M^{b}(d,k)=2((d1)^{k}2)(d2)^{1}. Bipartite (d,k)graphs whose order attains the bipartite Moore bound are called bipartite Moore graphs.
Bipartite Moore graphs have proved to be very rare. Feit and Higman, and also independently Singleton, proved that such graphs exist only when the diameter is 2,3,4 or 6. In the cases when the diameter is 3, 4 or 6, they have been constructed only when d1 is a prime power.
Therefore, in attempting to settle the values of N^{b}(d,k), research activities in this problem have follow the following two directions:
 Increasing the lower bounds for N^{b}(d,k) by constructing ever larger graphs.
 Lowering and/or setting upper bounds for N^{b}(d,k) by proving the nonexistence of graphs
whose order is close to the bipartite Moore bounds M^{b}(d,k)=2((d1)^{k}1)(d2)^{1}.
Increasing the lower bounds for N^{b}(d,k)
In recent years there has not been much activity in the constructions of large bipartite graphs. This may be, in part, because there was not an online table showing the latest constructions. In this direction Charles Delorme (in some cases collaborating with Bond and GómezMartí) provided some large bipartite graphs by using graph compounding, the concept of partial Cayley graph, and other techniques.
Now, with the release of this online table (see below), we expect to stimulate further research on this area.
Below is the table of the largest known bipartite graphs (as of January 2012) in the undirected degree diameter problem for bipartite graphs of degree at most 3 ≤ d ≤ 16 and diameter 3 ≤ k ≤ 10. This table represents the best lower bounds known at present on the order of (d,k)bipartite graphs. Many of the graphs of diameter 3 ,4 and 6 are bipartite Moore graphs, and thus are optimal. All optimal graphs are marked in bold.
Table of the orders of the largest known bipartite graphs
d\k  3  4  5  6  7  8  9  10

3  14  30  56  126  168  256  506  800

4  26  80  160  728  840  2 184  4 970  11 748

5  42  170  336  2 730  3 110  9 234  27 936  90 068

6  62  312  684  7 812  8 310  29 790  117 360  452 032

7  80  346  1 134  8 992  23 436  80 940  400 160  1 987 380

8  114  800  1 710  39 216  40 586  201 480  1 091 232  6 927 210

9  146  1 170  2 496  74 898  117 648  449 480  2 961 536  20 017 260

10  182  1 640  4 000  132 860  224 694  1 176 480  7 057 400  50 331 156

11  190  1 734  5 850  142 464  398 580  2 246 940  15 200 448  130 592 354

12  266  2 928  8 200  354 312  664 300  4 650 100  30 001 152  300 383 050

13  270  3 064  11 480  374 452  1 062 936  5 314 680  50 990 610  617 330 936

14  366  4 760  14 760  804 468  1 771 560  14 172 480  95 087 738  1 213 477 190

15  370  4 946  20 496  842 048  2 480 184  14 172 480  168 016 334  2 300 326 510

16  394  5 134  27 300  884 062  4 022 340  36 201 060  288 939 118  4 119 507 330

The following table is the key to the colors in the table presented above:
Color  Details

*  Bipartite Moore graphs (optimal).

*  Graph duplications found by C. Delorme and G. Farhi.

*  Graphs found by C. Delorme, J. Gómez, and J. J. Quisquater.

*  Optimal graph found by R. BarYehuda and T. Etzion and by J. Bond and C. Delorme.

*  Graph found independently by M. Conder and R. Nedela., by C. Delorme, J. Gómez, and J. J. Quisquater and by Eyal Loz.

*  Graphs found independently by Paul Hafner and by Eyal Loz.

*  Graphs found by Eyal Loz as part of the joint project The degree/diameter problem for several classes of graphs by E. Loz, H. PérezRosés and G. PinedaVillavicencio.

*  Graphs found by R. FeriaPuron, M. Miller and G. PinedaVillavicencio.

Lowering and/or setting upper bounds for N^{b}(d,k)
The Moore bound can be reached in some cases, but not always in general. Some theoretical work was done to determine the lowest upper bounds. In this direction reserachers have been interested in bipartite graphs of maximum degree d, diameter k and order M^{b}(d,k)δ for small δ. The parameter δ is called the defect. Such graphs are called bipartite (d,k,δ)graphs.
The bipartite (d,k,2;)graphs constitute the first interesting family of graphs to be studied. When d≥3 and k=2, bipartite (d,k,2)graphs are the complete bipartite graphs with partite sets of orders p and q, where either p=q=d1 or p=d and q=d2. For d≥3 and k≥3 only two such graphs are known; a unique bipartite (3, 3,2)graph and a unique bipartite (4, 3,2)graph.
Studies on bipartite (d,k,2;)graphs have been carried out by Charles Delorme, Leif Jorgensen, Mirka Miller and Guillermo PinedaVillavicencio. They proved several necessary conditions for the existence of bipartite (d,3,2;)graphs, the uniqueness of the two known bipartite (d,k,2;)graphs for d≥3 and k≥3, and the nonexistence of bipartite (d,k,2;)graphs for d≥3 and k≥4.
Lowest known upper bounds and the percentage of the order of the largest known bipartite graphs
d\k  3  4  5  6  7  8  9  10

3









4









5









6









7









8









9









10









11









12









13









14









15









16









The following table is the key to the colors in the table presented above:
Color  Details

*  Moore bound.

*  Moore bound minus 2.

*  Moore bound minus 6.

References
 Conder, M.; Nedela, R. (2006), "A more detailed classification of symmetric cubic graphs", preprint.
 BarYehuda, R.; Etzion, T. (1992), "Connections between two cycles  a new design of dense processor interconnection networks", Discrete Applied Mathematics 3738.
 Bond, J.; Delorme, C. (1988), "New large bipartite graphs with given degree and diameter", Ars Combinatoria 25C: 123132.
 Bond, J.; Delorme, C. (1993), "A note on partial Cayley graphs", Discrete Mathematics 114 (13): 6374, doi:10.1016/0012365X(93)90356X.
 Delorme, C. (1985), "Grands graphes de degré et diamètre donnés", European Journal of Combinatorics 6: 291302.
 Delorme, C. (1985), "Large bipartite graphs with given degree and diameter", Journal of Graph Theory 8: 325334.
 Delorme, C.; Farhi, G. (1984), "Large graphs with given degree and diameter Part I", IEEE Transactions on Computers C33: 857860.
 Delorme, C.; Gómez (2002), "Some new large compound graphs", European Journal of Combinatorics 23 (5): 539547, doi:10.1006/eujc.2002.0581.
 Delorme, C.; Gómez, J.; Quisquater, J. J., "On large bipartite graphs", submitted.
 Delorme, C.; Jorgensen, L.; Miller, M.; PinedaVillavicencio, G., "On bipartite graphs of diameter 3 and defect 2", Journal of Graph Theory 61 (2009), no. 4, 271288.
 Delorme, C.; Jorgensen, L.; Miller, M.; PinedaVillavicencio, G., "On bipartite graphs of defect 2", European Journal of Combinatorics 30 (2009), no. 4, 798808.
 PinedaVillavicencio, G., Nonexistence of bipartite graphs of diameter at least 4 and defect 2, Journal of Algebraic Combinatorics 34 (2011), no. 2, 163182.
 Miller, M.; Širáň, J. (2005), "Moore graphs and beyond: A survey of the degree/diameter problem", Electronic Journal of Combinatorics Dynamic survey D, PDF version.
External links
 Eyal Loz's DegreeDiameter problem page, including adjacency lists for bipartite graphs smaller than 20,000 found as a part of the project The degree/diameter problem for several classes of graphs.