# Difference between revisions of "The Degree Diameter Problem for General Graphs"

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## Introduction

The degree/diameter problem for general graphs can be stated as follows:

Given natural numbers d and k, find the largest possible number N(d,k) of vertices in a graph of maximum degree d and diameter k.

In attempting to settle the values of N(d,k), research activities in this problem have follow the following two directions:

• Increasing the lower bounds for N(d,k) by constructing ever larger graphs.
• Lowering and/or setting upper bounds for N(d,k) by proving the non-existence of graphs

whose order is close to the Moore bounds M(d,k)=(d(d-1)k-2)(d-2)-1.

## Increasing the lower bounds for N(d,k)

In the quest for the largest known graphs many innovative approaches have been suggested. In a wide spectrum, we can classify these approaches into general (those producing graphs for many combinations of the degree and the diameter) and ad hoc (those devised specifically for producing graphs for few combinations of the degree and the diameter). Among the former, we have the constructions of De Bruijn graphs and Kautz graphs, while among the latter, we have the star product, the voltage assigment technique and graph compunding. For information on the state-of -the-art of this research stream, the interested reader is referred to the survey by Miller and Širáň.

Below is the table of the largest known graphs (as of September 2009) in the undirected degree diameter problem for graphs of degree at most 3 ≤ d ≤ 20 and diameter 2 ≤ k ≤ 10. Only a few of the graphs in this table are known to be optimal (marked in bold), and thus, finding a larger graph that is closer in order (in terms of the size of the vertex set) to the Moore bound is considered an open problem. Some general constructions are known for values of d and k outside the range shown in the table.

### Table of the orders of the largest known graphs for the undirected degree diameter problem

 [math]d[/math]\[math]k[/math] 2 3 4 5 6 7 8 9 10 3 10 20 38 70 132 196 360 600 1 250 4 15 41 98 364 740 1 320 3 243 7 575 17 703 5 24 72 212 624 2 772 5 516 17 030 57 840 187 056 6 32 111 390 1 404 7 917 19 383 76 461 331 387 1 253 615 7 50 168 672 2 756 11 988 52 768 249 660 1 223 050 6 007 230 8 57 253 1 100 5 060 39 672 131 137 734 820 4 243 100 24 897 161 9 74 585 1 550 8 268 75 893 279 616 1 697 688 12 123 288 65 866 350 10 91 650 2 286 13 140 134 690 583 083 4 293 452 27 997 191 201 038 922 11 104 715 3 200 19 500 156 864 1 001 268 7 442 328 72 933 102 600 380 000 12 133 786 4 680 29 470 359 772 1 999 500 15 924 326 158 158 875 1 506 252 500 13 162 851 6 560 40 260 531 440 3 322 080 29 927 790 249 155 760 3 077 200 700 14 183 916 8 200 57 837 816 294 6 200 460 55 913 932 600 123 780 7 041 746 081 15 187 1 215 11 712 76 518 1 417 248 8 599 986 90 001 236 1 171 998 164 10 012 349 898 16 200 1 600 14 640 132 496 1 771 560 14 882 658 140 559 416 2 025 125 476 12 951 451 931 17 274 1 610 19 040 133 144 3 217 872 18 495 162 220 990 700 3 372 648 954 15 317 070 720 18 307 1 620 23 800 171 828 4 022 340 26 515 120 323 037 476 5 768 971 167 16 659 077 632 19 338 1 638 23 970 221 676 4 024 707 39 123 116 501 001 000 8 855 580 344 18 155 097 232 20 381 1 958 34 952 281 820 8 947 848 55 625 185 762 374 779 12 951 451 931 78 186 295 824

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

 Color Details * The Petersen and Hoffman–Singleton graphs. * Other non Moore but optimal graphs. * Graph found by J. Allwright. * Graph found by G. Wegner. * Graphs found by G. Exoo. * Family of graphs found by B. D. McKay, M. Miller and J. Širáň. More details are available in a paper by the authors. * Graphs found by J. Gómez. * Graph found by M. Mitjana and F. Comellas. This graph was also found independently by M. Sampels. * Graphs found by C. Delorme. * Graphs found by C. Delorme and G. Farhi. * Graphs found by E. Canale. (2012) * Graph found by J. C. Bermond, C. Delorme, and G. Farhi * Graphs found by J. Gómez and M. A. Fiol. * Graphs found by J. Gómez, M. A. Fiol, and O. Serra. * Graph found by M.A. Fiol and J.L.A. Yebra. * Graph found by F. Comellas and J. Gómez. * Graph found by Jianxiang Chen. * Graphs found by G. Pineda-Villavicencio, J. Gómez, M. Miller and H. Pérez-Rosés. More details are available in a paper by the authors. * Graphs found by E. Loz. More details are available in a paper by E. Loz and J. Širáň. * Graphs found by E. Loz and G. Pineda-Villavicencio. More details are available in a paper by the authors. * Graphs found by A. Rodriguez. (2012) * Graphs found by M. Sampels. * Graphs found by M. J. Dinneen and P. Hafner. More details are available in a paper by the authors. * Graph found by M. Conder. * Graphs found by Brown, W. G. (1966). * Graph found by M. Abas. (2017). More details are available in a paper by the author.

## Lowering and/or setting upper bounds for N(d,k)

As the Moore bound cannot be reached in general, some theoretical work has been done to determine the lowest possible upper bounds. In this direction reserachers have been interested in graphs of maximum degree d, diameter k and order M(d,k)-δ for small δ. The parameter δ is called the defect. Such graphs are called (d,k,-δ)-graphs.

For δ=1 the only (d,k,-1)-graphs are the cycles on 2k vertices. Erdös, Fajtlowitcz and Hoffman, who proved the non-existence of (d,2,-1)-graphs for d≠3. Then, Bannai and Ito, and also independently, Kurosawa and Tsujii, proved the non-existence of (d,k,-1)-graphs for d≥3 and k≥3.

For δ=2, the (2,k,-2)-graphs are the cycles on 2k-1. Considering d≥3, only five graphs are known at present. Elspas found the unique (4,2,-2)-graph and the unique (5,2,-2)-graph, and credited Green with producing the unique (3,3,-2)-graph. The other graphs are two non-isomorphic (3,2,-2)-graphs.

When δ=2, d≥3 and k≥3, not much is known about the existence or otherwise of (d,k,-2)-graphs. In this context some known outcomes include the non-existence of (3,k,-2)-graphs with k≥4 by Leif Jorgensen, the non-existence of (4,k,-2)-graphs with k≥3 by Mirka Miller and Rino Simanjuntak, some structural properties of (5,k,-2)-graphs with k≥3 by Guillermo Pineda-Villavicencio and Mirka Miller, the obtaining of several necessary conditions for the existence of (d,2,-2)-graphs with d≥3 by Mirka Miller, Minh Nguyen and Guillermo Pineda-Villavicencio, and the non-existence of (d,2,-2)-graphs for 5<d<50 by Jose Conde and Joan Gimbert.

For the case of δ≥3 only a few works are known at present: the non-existence of (3,4,-4)-graphs by Leif Jorgensen; the complete catalogue of (3,k,-4)-graphs with k≥2 by Guillermo Pineda-Villavicencio and Mirka Miller by proving the non-existence of (3,k,-4)-graphs with k≥5, the settlement of N(3,4)=M(3,4)=38 by Buset; and the obtaining of N(6,2)=M(6,2)-5=32 by Molodtsov. For more information, check the corresponding papers, and the survey by Miller and Širáň.

### Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs

[math]d[/math]\[math]k[/math] 2 3 4 5 6 7 8 9 10
3
 10 100%
 20 100%
 38 100%
 92 76.08%
 188 70.21%
 380 51.57%
 764 43.97%
 1532 39.16%
 3068 40.74%
4
 15 100%
 52 78.84%
 160 61.25%
 484 75.20%
 1456 50.82%
 4372 30.19%
 13120 24.71%
 39364 19.24%
 118096 14.99%
5
 24 100%
 104 69.23%
 424 50%
 1704 36.61%
 6824 40.62%
 27304 20.20%
 109224 15.59%
 436904 13.23%
 1747624 10.70%
6
 32 100%
 186 59.67%
 936 41.66%
 4686 29.96%
 23436 33.78%
 117186 16.54%
 585936 13.04%
 2929686 10.50%
 14648436 8.55%
7
 50 100%
 301 55.81%
 1813 37.06%
 10885 25.31%
 65317 18.35%
 391909 13.46%
 2351461 10.61%
 14108773 8.66%
 84652645 7.09%
8
 63 90.47%
 456 55.48%
 3200 34.37%
 22408 22.58%
 156864 25.29%
 1098056 11.94%
 7686400 9.56%
 53804808 7.88%
 376633664 6.61%
9
 80 92.50%
 657 89.04%
 5265 29.43%
 42129 19.46%
 337041 22.51%
 2696337 10.37%
 21570705 7.87%
 172565649 7.02%
 1380525201 4.77%
10
 99 91.91%
 910 71.42%
 8200 27.87%
 73810 17.80%
 664300 20.27%
 5978710 9.75%
 53808400 7.97%
 484275610 5.78%
 4358480500 4.61%
11
 120 86.66%
 1221 58.55%
 12221 26.18%
 122221 15.95%
 1222221 12.83%
 12222221 8.19%
 122222221 6.08%
 1222222221 5.96%
 12222222221 4.91%
12
 143 93%
 1596 49.24%
 17568 26.63%
 193260 15.24%
 2125872 16.92%
 23384604 8.55%
 257230656 6.19%
 2829537228 5.58%
 31124909520 4.83%
13
 168 96.42%
 2041 41.69%
 24505 26.77%
 294073 13.69%
 3528889 15.05%
 42346681 7.84%
 508160185 5.88%
 6097922233 4.08%
 73175066809 4.20%
14
 195 93.84%
 2562 35.75%
 33320 24.60%
 433174 13.35%
 5631276 14.49%
 73206602 8.46%
 951685840 5.87%
 12371915934 4.85%
 160834907156 4.37%
15
 224 83.03%
 3165 38.38%
 44325 26.42%
 620565 12.33%
 8687925 16.31%
 121630965 7.07%
 1702833525 5.28%
 23839669365 4.91%
 333755371125 2.99%
16
 255 77.64%
 3856 41.49%
 57856 25.30%
 867856 15.26%
 13017856 13.60%
 195267856 7.62%
 2929017856 4.79%
 43935267856 4.60%
 659029017856 1.96%
17
 288 95.13%
 4641 34.69%
 74273 25.63%
 1188385 11.20%
 19014177 16.92%
 304226849 6.07%
 4867629601 4.54%
 77882073633 4.33%
 1246113178145 1.22%
18
 323 95.04%
 5526 29.31%
 93960 25.32%
 1597338 10.75%
 27154764 14.81%
 461631006 5.74%
 7847727120 4.11%
 133411361058 4.32%
 2267993138004 0.73%
19
 360 93.88%
 6517 25.13%
 117325 20.43%
 2111869 10.49%
 38013661 10.58%
 684245917 5.71%
 12316426525 4.06%
 221695677469 3.99%
 3990522194461 0.45%
20
 399 95.48%
 7620 25.69%
 144800 24.13%
 2751220 10.24%
 52273200 17.11%
 993190820 5.6%
 18870625600 4.04%
 358541886420 3.61%
 6812295842000 1.14%

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

 Color Details * The Moore bound. * Upper bound introduced by A. Hoffman, R. Singleton, Bannai, E. and Ito, T. * Upper bound introduced by Leif Jorgensen. * Optimal graphs found by Buset and by Molodtsov. * Graphs shown optimal.

## References

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• Bannai, E.; Ito, T. (1981), "Regular graphs with excess one", Discrete Mathematics 37:147-158, doi:10.1016/0012-365X(81)90215-6.
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• J. Dinneen, Michael; Hafner, P. R. (1994), "New Results for the Degree/Diameter Problem", Networks 24 (7): 359–367, PDF version.
• Elspas, B. (1964), "Topological constraints on interconnection-limited logic", Proceedings of IEEE Fifth Symposium on Switching Circuit Theory and Logical Design S-164: 133--147.
• Erdös P; Fajtlowicz, S.; Hoffman A. J. (1980), "Maximum degree in graphs of diameter 2", Networks 10: 87-90.
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• Loz, E.; Širáň, J. (2008), "New record graphs in the degree-diameter problem", Australasian Journal of Combinatorics 41: 63–80.
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• Pineda-Villavicencio, G.; Miller, M. (Oct 2006), "On Graphs of Maximum Degree 5, Diameter D and Defect 2", Proceedings of MEMICS 2006, Second Doctoral Workshop on Mathematical and Engineering Methods in Computer Science: 182--189, Mikulov, Czech Republic.
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