Difference between revisions of "The Degree Diameter Problem for Circulant Graphs"

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(→‎Table of the orders of the largest known circulant graphs)
(→‎Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs)
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|35%
 
|35%
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|'''18'''
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| align="center" |
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{| border="2" style="background:rgb(180,180,255);"
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| 138
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|'''100%'''
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| 1 159
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|
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{| border="2" style="background:#ABCDEF;"
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| 5 641
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|47%
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{| border="2" style="background:#ABCDEF;"
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| 22 363
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|37%
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{| border="2" style="background:#ABCDEF;"
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| 75 517
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|36%
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{| border="2" style="background:#ABCDEF;"
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| 224 143
 +
|-
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|35%
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|}
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| align="center" |
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{| border="2" style="background:#ABCDEF;"
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| 598 417
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|-
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|34%
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|}
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| align="center" |
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{| border="2" style="background:#ABCDEF;"
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| 1 462 563
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|-
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|33%
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|}
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| align="center" |
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{| border="2" style="background:#ABCDEF;"
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| 3 317 445
 +
|-
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|31%
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|}
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| align="center" |
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{| border="2" style="background:#ABCDEF;"
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| 7 059 735
 +
|-
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|31%
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|}
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| align="center" |
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{| border="2" style="background:#ABCDEF;"
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| 14 218 905
 +
|-
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|31%
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|}
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|'''19'''
 +
| align="center" |
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{| border="2" style="background:rgb(180,180,255);"
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| 156
 +
|-
 +
|'''100%'''
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|}
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| align="center" |
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{| border="2" style="background:#ABCDEF;"
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| 1 340
 +
|-
 +
|
 +
|}
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| align="center" |
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{| border="2" style="background:#ABCDEF;"
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| 6 800
 +
|-
 +
|
 +
|}
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| align="center" |
 +
{| border="2" style="background:#ABCDEF;"
 +
| 28 004
 +
|-
 +
|34%
 +
|}
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| align="center" |
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{| border="2" style="background:#ABCDEF;"
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| 97 880
 +
|-
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|31%
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|}
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| align="center" |
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{| border="2" style="background:#ABCDEF;"
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| 299 660
 +
|-
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|33%
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|}
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| align="center" |
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{| border="2" style="background:#ABCDEF;"
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| 822 560
 +
|-
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|31%
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|}
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| align="center" |
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{| border="2" style="background:#ABCDEF;"
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| 2 060 980
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|-
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|32%
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|}
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| align="center" |
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{| border="2" style="background:#ABCDEF;"
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| 4 780 008
 +
|-
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|29%
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|}
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| align="center" |
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{| border="2" style="background:#ABCDEF;"
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| 10 377 180
 +
|-
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|30%
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|}
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| align="center" |
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{| border="2" style="background:#ABCDEF;"
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| 21 278 640
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|-
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|28%
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|}
 +
 +
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|'''20'''
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| align="center" |
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{| border="2" style="background:rgb(180,180,255);"
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| 171
 +
|-
 +
|'''100%'''
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|}
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| align="center" |
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{| border="2" style="background:#ABCDEF;"
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| 1 561
 +
|-
 +
|
 +
|}
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| align="center" |
 +
{| border="2" style="background:#ABCDEF;"
 +
| 8 361
 +
|-
 +
|34%
 +
|}
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| align="center" |
 +
{| border="2" style="background:#ABCDEF;"
 +
| 36 365
 +
|-
 +
|34%
 +
|}
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| align="center" |
 +
{| border="2" style="background:#ABCDEF;"
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| 134 245
 +
|-
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|30%
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|}
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| align="center" |
 +
{| border="2" style="background:#ABCDEF;"
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| 433 905
 +
|-
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|30%
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|}
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| align="center" |
 +
{| border="2" style="background:#ABCDEF;"
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| 1 256 465
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|-
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|28%
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|}
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| align="center" |
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{| border="2" style="background:#ABCDEF;"
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| 3 317 445
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|-
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|28%
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|}
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| align="center" |
 +
{| border="2" style="background:#ABCDEF;"
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| 8 097 453
 +
|-
 +
|28%
 +
|}
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| align="center" |
 +
{| border="2" style="background:#ABCDEF;"
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| 18 474 633
 +
|-
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|25%
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|}
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| align="center" |
 +
{| border="2" style="background:#ABCDEF;"
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| 39 753 273
 +
|-
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|25%
 
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Revision as of 14:43, 7 June 2019

Table of the orders of the largest known circulant graphs

[math]d[/math]\[math]k[/math] 2 3 4 5 6 7 8 9 10 11 12
3 8 12 16 20 24 28 32 36 40 44 48
4 13 25 41 61 85 113 145 181 221 265 313
5 16 36 64 100 144 196 256 324 400 484 576
6 21 55 117 203 333 515 737 1 027 1 393 1 815 2 329
7 26 76 160 308 536 828 1 232 1 764 2 392 3 180 4 144
8 35 104 248 528 984 1 712 2 768 4 280 6 320 9 048 12 552
9 42 130 320 700 1 416 2 548 4 304 6 804 10 320 15 004 21 192
10 51 177 457 1 099 2 380 4 551 8 288 14 099 22 805 35 568 53 025
11 56 210 576 1 428 3 200 6 652 12 416 21 572 35 880 56 700 87 248
12 67 275 819 2 120 5 044 10 777 21 384 39 996 69 965 117 712 190 392
13 80 312 970 2 676 6 256 14 740 30 760 57 396 106 120 182 980 295 840
14 90 381 1 229 3 695 9 800 23 304 49 757 103 380 196 689 350 700 593 989
15 96 448 1 420 4 292 12 232 32 092 68 944 142 516 276 928 514 580 908 480
16 112 518 1 788 5 847 17 733 44 328 107 748 232 245 479 255 924 420 1 702 428
17 130 570 1 954 6 468 20 360 57 684 136 512 321 780 659 464 1 350 820 2 479 104
18 138 2 645 8 248 27 273 77 577 205 601 483 523 1 024 915 2 202 955 4 388 640
19 156 9 652 29 928 99 420 257 144 652 004 1 388 608 3 101 860 6 037 496
20 171 2 828 12 396 39 733 131 835 358 089 930 184 2 232 648 4 529 265 10 001 820

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

Color Details
* Numbers in bold indicate graphs known to be optimal.
* Optimal graphs.
* Optimal graphs found by E. Monakhova.
* Graphs found by H. Macbeth, J. Šiagiová, J. Širáň and T. Vetrík.
* Graphs found by R. Dougherty and V. Faber and independently for d=6 by E. Monakhova.
* Graphs found by B. McKay.
* Graphs found by R. Lewis.
* Graphs found by R. Lewis and independently by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
* Graphs found by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
* Graphs found by D. Bevan, G. Erskine and R. Lewis.
* Graphs found by G. Erskine.
* Graphs found by O. Monakhov and E. Monakhova.

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 11 12
3
8
100%
12
100%
16
100%
20
100%
24
100%
28
100%
32
100%
36
100%
44
100%
48
100%
40
100%
4
13
100%
25
100%
41
100%
61
100%
85
100%
113
100%
145
100%
181
100%
221
100%
265
100%
313
100%
5
16
100%
36
100%
64
100%
100
100%
144
100%
196
100%
256
100%
324
100%
400
100%
484
100%
576
100%
6
21
100%
55
100%
117
100%
203
100%
333
100%
515
100%
737
100%
1 027
100%
1 393
100%
1 815
100%
2 329
100%
7
26
100%
76
100%
160
100%
308
100%
536
100%
828
100%
1 232
100%
1 764
100%
2 392
100%
3 608
88%
4 672
89%
8
35
100%
104
100%
248
100%
528
100%
984
100%
1 712
100%
3 649
76%
5 641
76%
8 361
76%
11 969
76%
16 641
75%
9
42
100%
130
100%
320
100%
700
100%
1 416
100%
3 530
72%
5 890
73%
9 290
73%
14 002
74%
20 330
74%
28 610
74%
10
51
100%
177
100%
681
67%
1 683
65%
3 653
65%
7 183
63%
13 073
63%
22 363
63%
36 365
63%
56 695
63%
85 305
62%
11
56
100%
210
100%
912
63%
2 364
60%
5 336
60%
10 836
61%
20 256
61%
35 436
61%
58 728
61%
93 060
61%
142 000
61%
12
67
100%
275
100%
1 289
64%
3 653
58%
8 989
56%
19 825
54%
40 081
53%
75 517
53%
134 245
52%
227 305
52%
369 305
52%
13
80
100%
312
100%
1 666
58%
4 942
54%
12 642
49%
28 814
51%
59 906
51%
115 598
50%
209 762
51%
361 550
51%
596610
50%
14
90
100%
381
100%
2 241
55%
7 183
51%
19 825
49%
48 639
48%
108 545
46%
224 143
46%
433 905
45%
795 455
44%
1 392 065
43%
15
96
100%
448
100%
2 816
50%
9 424
46%
27 008
45%
68 464
47%
157 184
44%
332 688
43%
658 048
42%
1 229 360
42%
2 187 520
42%
16
112
100%
833
62%
3 649
49%
13 073
45%
40 081
44%
108 545
41%
265 729
41%
598 417
39%
1 256 465
38%
2 485 825
37%
4 673 345
36%
17
130
100%
978
58%
4 482
44%
16 722
39%
53 154
38%
148 626
39%
374 274
36%
864 146
37%
1 854 882
36%
3 742 290
36%
7 159 170
35%


18
138
100%
1 159
5 641
47%
22 363
37%
75 517
36%
224 143
35%
598 417
34%
1 462 563
33%
3 317 445
31%
7 059 735
31%
14 218 905
31%


19
156
100%
1 340
6 800
28 004
34%
97 880
31%
299 660
33%
822 560
31%
2 060 980
32%
4 780 008
29%
10 377 180
30%
21 278 640
28%


20
171
100%
1 561
8 361
34%
36 365
34%
134 245
30%
433 905
30%
1 256 465
28%
3 317 445
28%
8 097 453
28%
18 474 633
25%
39 753 273
25%

References

  • D. Bevan, G. Erskine, and R. Lewis. Large circulant graphs of fixed diameter and arbitrary degree. ArXiv
  • R. Feria-Puron, J. Ryan, and H. Perez-Roses. Searching for Large Multi-Loop Networks. Electronic Notes in Discrete Mathematics, vol. 46 (2014), pp. 233-240. doi:10.1016/j.endm.2014.08.031. Link to journal
  • R.R. Lewis. The Degree/Diameter Problem for Circulant Graphs of Degree 8 and 9. The Electronic Journal of Combinatorics, vol. 21(4) (2014), #P4.50. Link to journal
  • E.A. Monakhova, Synthesis of optimal Diophantine structures, Comput. Syst. Novosibirsk , 80 (1979), p.18--35. (in Russian).
  • E. Monakhova, Optimal Triple Loop Networks with Given Transmission Delay: Topological Design and Routing, Inter. Network Optimization Conference, (INOC'2003), Evry/Paris, France, (2003), p.410--415.
  • E.A. Monakhova . On synthesis of multidimensional circulant graphs of diameter two, Bulletin of the Tomsk Polytechnic University. 323(2) (2013), p.25--28. (in Russian). Link to journal
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