Numbers of characteristic polynomials and cospectral graphs for A

Number of distinct characteristic polynomials, and number of graphs with a cospectral mate for the adjacency matrix A.

n #graphs #char. pols #graphs with mate

0 1 1 0

1 1 1 0

2 2 2 0

3 4 4 0

4 11 11 0

5 34 33 2

6 156 151 10

7 1044 988 110

8 12346 11453 1722

9 274668 247357 51039*

10 12005168 10608128 2560606*

11 1018997864 901029366 215331676*

12 165091172592 148187993520 31067572481

n	#graphs	#char. pols	#graphs with mate
0	1	1	0
1	1	1	0
2	2	2	0
3	4	4	0
4	11	11	0
5	34	33	2
6	156	151	10
7	1044	988	110
8	12346	11453	1722
9	274668	247357	51039*
10	12005168	10608128	2560606*
11	1018997864	901029366	215331676*
12	165091172592	148187993520	31067572481

The three starred entries are 1 more, 90 more, and 1 less than the corresponding values that Haemers & Spence found. The first entry is confirmed by Godsil & McKay (who gave data up to n=9). Lepović has collected statistical data on connected graphs on 10 vertices, so his data is not directly comparable to the data here. The entire table above has now been confirmed by Spence.

References

C. Godsil & B. McKay, Some computational results on the spectra of graphs, in: Combinatorial Mathematics IV, Springer LNM 560 (1976) 73-92.

W. H. Haemers & E. Spence, Enumeration of cospectral graphs, Europ. J. Combin. 25 (2004) 199-211.

Mirko Lepović, Some statistical data on graphs with 10 vertices, Univ. Beograd Publikacija Elektrotehničkog Fakulteta - Serija: Matematika 9 (1998) 79-88.

The number of distinct characteristic polynomials for simple graphs on n vertices is sequence A082104 in the Encyclopedia of Integer Sequences.

Number of graphs per polynomial

Number of characteristic polynomials that occur for m graphs, for the adjacency matrix A.

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 21

5 32 1

6 146 5

7 934 52 2

8 10624 771 52 6

9 223629 21025 2015 551 95 37 1 2 - 2

10 9444562 1003418 110812 36748 5672 4693 948 870 125 142 72 28 9 13 10 4 2

For n=11 the largest family of cospectral graphs has size 46. For n=12 it has size 128.

Subdivided according to number of edges

Below the same data again, subdivided according to number of edges. Perhaps one is mostly interested in the totals, but because the files are large, and the spectrum determines the number of edges and of triangles, it is easiest to subdivide the generation process. As a side result we have counts of the number of simple graphs with given numbers of vertices, edges, and triangles.)

n=5 n=6 n=7 n=8 n=9 n=10 n=11 n=12

n < 5

No cospectral graphs.

n=5

Here and below, e gives the numer of edges, #cp gives the number of distinct characteristic polynomials, #cpN (for N=1,2,...) gives the number of characteristic polynomials that occur for precisely N graphs.

e #graphs #cp #cp1 #cp2 #with mate

0 1 1 1 - 0

1 1 1 1 - 0

2 2 2 2 - 0

3 4 4 4 - 0

4 6 5 4 1 2

5 6 6 6 - 0

6 6 6 6 - 0

7 4 4 4 - 0

8 2 2 2 - 0

9 1 1 1 - 0

10 1 1 1 - 0

Here a single cospectral pair: K₁+C₄ and K_1,4, with spectrum 2,0,0,0,–2.

n=6

e	#graphs	#cp	#cp1	#cp2	#with mate
0	1	1	1	-	0
1	1	1	1	-	0
2	2	2	2	-	0
3	5	5	5	-	0
4	9	7	5	2	4
5	15	14	13	1	2
6	21	20	19	1	2
7	24	23	22	1	2
8	24	24	24	-	0
9	21	21	21	-	0
10	15	15	15	-	0
11	9	9	9	-	0
12	5	5	5	-	0
13	2	2	2	-	0
14	1	1	1	-	0
15	1	1	1	-	0

n=7

e	#graphs	#cp	#cp1	#cp2	#cp3	#with mate
0	1	1	1	-	-	0
1	1	1	1	-	-	0
2	2	2	2	-	-	0
3	5	5	5	-	-	0
4	10	8	6	2	-	4
5	21	19	18	-	1	3
6	41	34	27	7	-	14
7	65	61	57	4	-	8
8	97	88	79	9	-	18
9	131	125	119	6	-	12
10	148	140	132	8	-	16
11	148	141	135	5	1	13
12	131	125	119	6	-	12
13	97	95	93	2	-	4
14	65	64	63	1	-	2
15	41	40	39	1	-	2
16	21	20	19	1	-	2
17	10	10	10	-	-	0
18	5	5	5	-	-	0
19	2	2	2	-	-	0
20	1	1	1	-	-	0
21	1	1	1	-	-	0

Here two triples of cospectral graphs, one with 5 and one with 11 edges. The first triple has spectrum 2, 1, 0, 0, 0, –1, –2 and consists of the three graphs K₁+K₂+C₄ and K_1,4+K₂ and K_1,2~K_1,2. The second triple has spectrum 3.49, 1.29, 0, 0, –1, –1.78, –2 (where the three nonintegers are roots of x³–3x²–4x+8 = 0). The adjacency matrices are

...1..1   ...1111   ..1.111
....111   ...1...   ...1.1.
....111   ....111   1...111
1.....1   11...1.   .1....1
.11...1   1.1...1   1.1....
.11...1   1.11..1   111...1
111111.   1.1.11.   1.11.1.

n=8

e	#graphs	#cp	#cp1	#cp2	#cp3	#cp4	#with mate
0	1	1	1	-	-	-	0
1	1	1	1	-	-	-	0
2	2	2	2	-	-	-	0
3	5	5	5	-	-	-	0
4	11	9	7	2	-	-	4
5	24	21	19	1	1	-	5
6	56	44	35	7	1	1	21
7	115	102	89	13	-	-	26
8	221	187	158	25	3	1	63
9	402	366	334	28	4	-	68
10	663	576	499	67	10	-	164
11	980	903	832	66	4	1	148
12	1312	1198	1093	97	7	1	219
13	1557	1443	1334	105	3	1	223
14	1646	1533	1427	99	7	-	219
15	1557	1451	1347	102	2	-	210
16	1312	1234	1161	68	5	-	151
17	980	933	889	42	1	1	91
18	663	630	599	29	2	-	64
19	402	387	372	15	-	-	30
20	221	215	211	2	2	-	10
21	115	113	111	2	-	-	4
22	56	55	54	1	-	-	2
23	24	24	24	-	-	-	0
24	11	11	11	-	-	-	0
25	5	5	5	-	-	-	0
26	2	2	2	-	-	-	0
27	1	1	1	-	-	-	0
28	1	1	1	-	-	-	0

n=9

e	#graphs	#cp	#cp1	#cp2	#cp3	#cp4	#cp5	#cp6	#cp7	#cp8	#cp9	#cp10	#with mate
0	1	1	1	-	-	-	-	-	-	-	-	-	0
1	1	1	1	-	-	-	-	-	-	-	-	-	0
2	2	2	2	-	-	-	-	-	-	-	-	-	0
3	5	5	5	-	-	-	-	-	-	-	-	-	0
4	11	9	7	2	-	-	-	-	-	-	-	-	4
5	25	22	20	1	1	-	-	-	-	-	-	-	5
6	63	49	39	8	1	-	1	-	-	-	-	-	24
7	148	120	97	19	3	1	-	-	-	-	-	-	51
8	345	269	209	50	5	4	1	-	-	-	-	-	136
9	771	658	567	75	11	4	1	-	-	-	-	-	204
10	1637	1347	1125	169	41	10	1	1	-	-	-	-	512
11	3252	2858	2512	305	35	5	1	-	-	-	-	-	740
12	5995	5203	4576	494	109	19	2	3	-	-	-	-	1419
13	10120	9012	8055	840	89	22	6	-	-	-	-	-	2065
14	15615	13826	12333	1273	167	37	10	5	1	-	-	-	3282
15	21933	19637	17602	1833	149	49	2	2	-	-	-	-	4331
16	27987	24993	22474	2173	261	57	18	8	-	1	-	1	5513
17	32403	29064	26065	2742	179	75	1	2	-	-	-	-	6338
18	34040	30602	27636	2615	257	72	17	5	-	-	-	-	6404
19	32403	29261	26413	2626	154	66	-	2	-	-	-	-	5990
20	27987	25320	23056	1979	205	54	20	4	-	1	-	1	4931*
21	21933	19988	18238	1600	110	36	3	1	-	-	-	-	3695
22	15615	14289	13157	983	118	20	8	3	-	-	-	-	2458
23	10120	9331	8620	652	41	17	1	-	-	-	-	-	1500
24	5995	5565	5206	296	58	3	1	1	-	-	-	-	789
25	3252	3050	2857	186	6	-	1	-	-	-	-	-	395
26	1637	1550	1475	63	12	-	-	-	-	-	-	-	162
27	771	739	708	30	1	-	-	-	-	-	-	-	63
28	345	335	327	6	2	-	-	-	-	-	-	-	18
29	148	144	140	4	-	-	-	-	-	-	-	-	8
30	63	62	61	1	-	-	-	-	-	-	-	-	2
31	25	25	25	-	-	-	-	-	-	-	-	-	0
32	11	11	11	-	-	-	-	-	-	-	-	-	0
33	5	5	5	-	-	-	-	-	-	-	-	-	0
34	2	2	2	-	-	-	-	-	-	-	-	-	0
35	1	1	1	-	-	-	-	-	-	-	-	-	0
36	1	1	1	-	-	-	-	-	-	-	-	-	0

The above data agrees with Godsil & McKay. For the starred entry Haemers & Spence gave 4930 instead of 4931.

n=10

In order not to make the following table too wide, the data for a given number of edges is given in several rows, with cp1-cp10 in the first row, cp11-cp20 in the second row, cp21-cp30 in the third row, etc.

e #graphs #cp #cp1 #cp2 #cp3 #cp4 #cp5 #cp6 #cp7 #cp8 #cp9 #cp10 #with mate

0 1 1 1 - - - - - - - - - 0

1 1 1 1 - - - - - - - - - 0

2 2 2 2 - - - - - - - - - 0

3 5 5 5 - - - - - - - - - 0

4 11 9 7 2 - - - - - - - - 4

5 26 23 21 1 1 - - - - - - - 5

6 66 51 40 9 1 - 1 - - - - - 26

7 165 130 103 22 3 1 1 - - - - - 62

8 428 316 237 61 9 3 6 - - - - - 191

9 1103 873 691 148 23 8 3 - - - - - 412

10 2769 2144 1701 326 77 24 11 2 2 1 - - 1068

11 6759 5655 4765 733 116 29 9 2 1 - - - 1994

12 15772 12992 10929 1583 332 94 30 16 6 1 1 - 4843

13 34663 29774 25789 3323 504 106 30 16 3 2 1 - 8874

14 71318 60763 52570 6549 1191 298 94 36 13 5 3 3 18748*

1 - - - - - - - - -

15 136433 118922 104581 12108 1618 415 121 55 13 7 2 1 31852

- 1 - - - - - - - -

16 241577 209938 184750 20773 3134 864 233 106 39 27 3 3 56827

2 1 1 1 1 - - - - -

17 395166 346777 307180 33616 4147 1304 254 183 49 28 7 2 87986

5 2 - - - - - - - -

18 596191 522518 462841 50164 6611 2074 385 292 60 58 12 11 133350

8 - 2 - - - - - - -

19 828728 729637 647492 70553 7978 2662 453 356 56 63 7 6 181236

7 2 1 1 - - - - - -

20 1061159 933585 827909 90921 10058 3466 544 443 93 99 13 23 233250

6 6 - 2 1 - - - - -

1 - - - - - - - - -

21 1251389 1102637 978053 108195 11105 4013 539 494 109 83 14 18 273336

3 2 2 3 2 2 - - - -

22 1358852 1198888 1064453 117027 11756 4363 542 510 107 100 12 8 294399

6 2 1 - 1 - - - - -

23 1358852 1200673 1067461 116158 11545 4269 508 499 103 98 13 10 291391

5 2 - 1 1 - - - - -

24 1251389 1106818 985095 106098 10662 3796 495 474 82 78 8 19 266294

2 2 1 2 2 2 - - - -

25 1061159 940404 839500 87450 9233 3141 461 401 81 91 11 20 221659

5 6 - 2 1 - - - - -

1 - - - - - - - - -

26 828728 737004 660011 66742 7183 2310 327 315 47 50 7 6 168717

5 1 - - - - - - - -

27 596191 531474 477924 45798 5522 1616 266 244 39 40 7 9 118267

8 - 1 - - - - - - -

28 395166 353751 319073 29898 3455 985 156 134 22 15 4 2 76093

6 1 - - - - - - - -

29 241577 217134 196949 17121 2297 533 122 74 19 13 - 1 44628

3 - - 1 1 - - - - -

30 136433 123348 112145 9755 1147 217 48 28 3 5 - - 24288

31 71318 64836 59390 4625 679 101 22 12 1 6 - - 11928

32 34663 31776 29231 2256 245 36 7 1 - - - - 5432

33 15772 14591 13584 857 129 18 3 - - - - - 2188

34 6759 6322 5919 372 29 1 1 - - - - - 840

35 2769 2615 2479 119 16 1 - - - - - - 290

36 1103 1054 1009 41 4 - - - - - - - 94

37 428 414 402 10 2 - - - - - - - 26

38 165 162 159 3 - - - - - - - - 6

39 66 65 64 1 - - - - - - - - 2

40 26 26 26 - - - - - - - - - 0

41 11 11 11 - - - - - - - - - 0

42 5 5 5 - - - - - - - - - 0

43 2 2 2 - - - - - - - - - 0

44 1 1 1 - - - - - - - - - 0

45 1 1 1 - - - - - - - - - 0

This data agrees with the data given by Haemers & Spence, except for the starred entry 18748 where they have 18747. That means that their total is 2560605, and that their 2560516 was a typo.

n=11

Here the last column gives the maximum size of a family of cospectral graphs.

e #graphs #cp #with mate maxsz

0 1 1 0 1

1 1 1 0 1

2 2 2 0 1

3 5 5 0 1

4 11 9 4 2

5 26 23 5 3

6 67 52 26 5

7 172 135 65 5

8 467 336 220 6

9 1305 980 558 6

10 3664 2685 1617 9

11 10250 8181 3601 8

12 28259 22304 10088 10

13 75415 63021 21915 11

14 192788 160048 56851 16

15 467807 401830 118099 21

16 1069890 918366 269166 21

17 2295898 2001891 529579 16

18 4609179 4022395 1054698 21

19 8640134 7593806 1892068* 20

20 15108047 13287522 3292566 31

21 24630887 21719491 5279190 30

22 37433760 33026554 8001477 39

23 53037356 46846429 11264629 35

24 70065437 61889850 14903754 36

25 86318670 76256313 18373280 31

26 99187806 87644859 21108349 46

27 106321628 93992968 22561018 34

28 106321628 94018562 22519077 34

29 99187806 87724533 20975573 46

30 86318670 76375210 18177143 31

31 70065437 62043591 14648996 36

32 53037356 47004809 11003951 34

33 37433760 33196606 7721507 33

34 24630887 21862432 5043664 30

35 15108047 13426080 3065242 30

36 8640134 7690246 1733808 20

37 4609179 4109009 913741 18

38 2295898 2052334 446687 16

39 1069890 959426 202832 15

40 467807 421573 85509 9

41 192788 174815 33221 9

42 75415 68958 12079 6

43 28259 26088 4048 5

44 10250 9582 1272 5

45 3664 3468 369 4

46 1305 1253 100 3

47 467 454 24 3

48 172 168 8 2

49 67 66 2 2

50 26 26 0 1

51 11 11 0 1

52 5 5 0 1

53 2 2 0 1

54 1 1 0 1

55 1 1 0 1

The starred entry is 1 less than the corresponding entry in Haemers & Spence. (And correspondingly our total is 1 less than their total.)

n=12

e	#graphs	#cp	#with mate	maxsz
0	1	1	0	1
1	1	1	0	1
2	2	2	0	1
3	5	5	0	1
4	11	9	4	2
5	26	23	5	3
6	68	53	26	5
7	175	137	67	5
8	485	346	231	7
9	1405	1028	630	8
10	4191	2930	2038	10
11	12763	9621	5277	10
12	39243	29405	16193	14
13	119890	96137	40764	15
14	359307	287874	121006	21
15	1043774	876408	293842	24
16	2911086	2451075	802394	26
17	7739601	6685738	1878695	20
18	19515361	16911773	4633795	31
19	46505609	40829479	10226536	36
20	104504341	92009161	22529245	47
21	221147351	196003994	45623540	42
22	440393606	391210005	89333350	58
23	825075506	735607356	163077317	75
24	1454265734	1298678301	283881928	84
25	2411961516	2158074983	464168797	70
26	3765262970	3372225877	719283179	88
27	5534255092	4962042192	1048804378	77
28	7661345277	6873359123	1445744465	98
29	9992340187	8969183184	1879609773	87
30	12281841209	11026119614	2308811620	89
31	14229503560	12777398409	2672120188	106
32	15542350436	13959109693	2914822990	98
33	16006173014	14377992933	2998429426	128
34	15542350436	13960847673	2912011113	98
35	14229503560	12780531627	2667028735	106
36	12281841209	11030772434	2301299608	89
37	9992340187	8974361437	1871227811	92
38	7661345277	6879443698	1435979682	98
39	5534255092	4967640411	1039800439	77
40	3765262970	3378040813	710017278	88
41	2411961516	2162716257	456756771	70
42	1454265734	1303076951	276928481	84
43	825075506	738656475	158242526	75
44	440393606	393882268	85138580	52
45	221147351	197621776	43066062	42
46	104504341	93316379	20482028	37
47	46505609	41511313	9147293	26
48	19515361	17422116	3836007	23
49	7739601	6918134	1508450	17
50	2911086	2608210	557209	15
51	1043774	939397	192946	10
52	359307	325447	62764	10
53	119890	109562	19299	8
54	39243	36247	5598	6
55	12763	11945	1555	5
56	4191	3974	407	4
57	1405	1350	106	3
58	485	471	26	3
59	175	172	6	2
60	68	67	2	2
61	26	26	0	1
62	11	11	0	1
63	5	5	0	1
64	2	2	0	1
65	1	1	0	1
66	1	1	0	1

n	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	21
5	32	1
6	146	5
7	934	52	2
8	10624	771	52	6
9	223629	21025	2015	551	95	37	1	2	-	2
10	9444562	1003418	110812	36748	5672	4693	948	870	125	142	72	28	9	13	10	4	2

e	#graphs	#cp	#cp1	#cp2	#with mate
0	1	1	1	-	0
1	1	1	1	-	0
2	2	2	2	-	0
3	4	4	4	-	0
4	6	5	4	1	2
5	6	6	6	-	0
6	6	6	6	-	0
7	4	4	4	-	0
8	2	2	2	-	0
9	1	1	1	-	0
10	1	1	1	-	0

e	#graphs	#cp	#cp1	#cp2	#with mate
0	1	1	1	-	0
1	1	1	1	-	0
2	2	2	2	-	0
3	5	5	5	-	0
4	9	7	5	2	4
5	15	14	13	1	2
6	21	20	19	1	2
7	24	23	22	1	2
8	24	24	24	-	0
9	21	21	21	-	0
10	15	15	15	-	0
11	9	9	9	-	0
12	5	5	5	-	0
13	2	2	2	-	0
14	1	1	1	-	0
15	1	1	1	-	0

e	#graphs	#cp	#cp1	#cp2	#cp3	#with mate
0	1	1	1	-	-	0
1	1	1	1	-	-	0
2	2	2	2	-	-	0
3	5	5	5	-	-	0
4	10	8	6	2	-	4
5	21	19	18	-	1	3
6	41	34	27	7	-	14
7	65	61	57	4	-	8
8	97	88	79	9	-	18
9	131	125	119	6	-	12
10	148	140	132	8	-	16
11	148	141	135	5	1	13
12	131	125	119	6	-	12
13	97	95	93	2	-	4
14	65	64	63	1	-	2
15	41	40	39	1	-	2
16	21	20	19	1	-	2
17	10	10	10	-	-	0
18	5	5	5	-	-	0
19	2	2	2	-	-	0
20	1	1	1	-	-	0
21	1	1	1	-	-	0

e	#graphs	#cp	#cp1	#cp2	#cp3	#cp4	#cp5	#cp6	#cp7	#cp8	#cp9	#cp10	#with mate
0	1	1	1	-	-	-	-	-	-	-	-	-	0
1	1	1	1	-	-	-	-	-	-	-	-	-	0
2	2	2	2	-	-	-	-	-	-	-	-	-	0
3	5	5	5	-	-	-	-	-	-	-	-	-	0
4	11	9	7	2	-	-	-	-	-	-	-	-	4
5	26	23	21	1	1	-	-	-	-	-	-	-	5
6	66	51	40	9	1	-	1	-	-	-	-	-	26
7	165	130	103	22	3	1	1	-	-	-	-	-	62
8	428	316	237	61	9	3	6	-	-	-	-	-	191
9	1103	873	691	148	23	8	3	-	-	-	-	-	412
10	2769	2144	1701	326	77	24	11	2	2	1	-	-	1068
11	6759	5655	4765	733	116	29	9	2	1	-	-	-	1994
12	15772	12992	10929	1583	332	94	30	16	6	1	1	-	4843
13	34663	29774	25789	3323	504	106	30	16	3	2	1	-	8874
14	71318	60763	52570	6549	1191	298	94	36	13	5	3	3	18748*
			1	-	-	-	-	-	-	-	-	-
15	136433	118922	104581	12108	1618	415	121	55	13	7	2	1	31852
			-	1	-	-	-	-	-	-	-	-
16	241577	209938	184750	20773	3134	864	233	106	39	27	3	3	56827
			2	1	1	1	1	-	-	-	-	-
17	395166	346777	307180	33616	4147	1304	254	183	49	28	7	2	87986
			5	2	-	-	-	-	-	-	-	-
18	596191	522518	462841	50164	6611	2074	385	292	60	58	12	11	133350
			8	-	2	-	-	-	-	-	-	-
19	828728	729637	647492	70553	7978	2662	453	356	56	63	7	6	181236
			7	2	1	1	-	-	-	-	-	-
20	1061159	933585	827909	90921	10058	3466	544	443	93	99	13	23	233250
			6	6	-	2	1	-	-	-	-	-
			1	-	-	-	-	-	-	-	-	-
21	1251389	1102637	978053	108195	11105	4013	539	494	109	83	14	18	273336
			3	2	2	3	2	2	-	-	-	-
22	1358852	1198888	1064453	117027	11756	4363	542	510	107	100	12	8	294399
			6	2	1	-	1	-	-	-	-	-
23	1358852	1200673	1067461	116158	11545	4269	508	499	103	98	13	10	291391
			5	2	-	1	1	-	-	-	-	-
24	1251389	1106818	985095	106098	10662	3796	495	474	82	78	8	19	266294
			2	2	1	2	2	2	-	-	-	-
25	1061159	940404	839500	87450	9233	3141	461	401	81	91	11	20	221659
			5	6	-	2	1	-	-	-	-	-
			1	-	-	-	-	-	-	-	-	-
26	828728	737004	660011	66742	7183	2310	327	315	47	50	7	6	168717
			5	1	-	-	-	-	-	-	-	-
27	596191	531474	477924	45798	5522	1616	266	244	39	40	7	9	118267
			8	-	1	-	-	-	-	-	-	-
28	395166	353751	319073	29898	3455	985	156	134	22	15	4	2	76093
			6	1	-	-	-	-	-	-	-	-
29	241577	217134	196949	17121	2297	533	122	74	19	13	-	1	44628
			3	-	-	1	1	-	-	-	-	-
30	136433	123348	112145	9755	1147	217	48	28	3	5	-	-	24288
31	71318	64836	59390	4625	679	101	22	12	1	6	-	-	11928
32	34663	31776	29231	2256	245	36	7	1	-	-	-	-	5432
33	15772	14591	13584	857	129	18	3	-	-	-	-	-	2188
34	6759	6322	5919	372	29	1	1	-	-	-	-	-	840
35	2769	2615	2479	119	16	1	-	-	-	-	-	-	290
36	1103	1054	1009	41	4	-	-	-	-	-	-	-	94
37	428	414	402	10	2	-	-	-	-	-	-	-	26
38	165	162	159	3	-	-	-	-	-	-	-	-	6
39	66	65	64	1	-	-	-	-	-	-	-	-	2
40	26	26	26	-	-	-	-	-	-	-	-	-	0
41	11	11	11	-	-	-	-	-	-	-	-	-	0
42	5	5	5	-	-	-	-	-	-	-	-	-	0
43	2	2	2	-	-	-	-	-	-	-	-	-	0
44	1	1	1	-	-	-	-	-	-	-	-	-	0
45	1	1	1	-	-	-	-	-	-	-	-	-	0

e	#graphs	#cp	#with mate	maxsz
0	1	1	0	1
1	1	1	0	1
2	2	2	0	1
3	5	5	0	1
4	11	9	4	2
5	26	23	5	3
6	67	52	26	5
7	172	135	65	5
8	467	336	220	6
9	1305	980	558	6
10	3664	2685	1617	9
11	10250	8181	3601	8
12	28259	22304	10088	10
13	75415	63021	21915	11
14	192788	160048	56851	16
15	467807	401830	118099	21
16	1069890	918366	269166	21
17	2295898	2001891	529579	16
18	4609179	4022395	1054698	21
19	8640134	7593806	1892068*	20
20	15108047	13287522	3292566	31
21	24630887	21719491	5279190	30
22	37433760	33026554	8001477	39
23	53037356	46846429	11264629	35
24	70065437	61889850	14903754	36
25	86318670	76256313	18373280	31
26	99187806	87644859	21108349	46
27	106321628	93992968	22561018	34
28	106321628	94018562	22519077	34
29	99187806	87724533	20975573	46
30	86318670	76375210	18177143	31
31	70065437	62043591	14648996	36
32	53037356	47004809	11003951	34
33	37433760	33196606	7721507	33
34	24630887	21862432	5043664	30
35	15108047	13426080	3065242	30
36	8640134	7690246	1733808	20
37	4609179	4109009	913741	18
38	2295898	2052334	446687	16
39	1069890	959426	202832	15
40	467807	421573	85509	9
41	192788	174815	33221	9
42	75415	68958	12079	6
43	28259	26088	4048	5
44	10250	9582	1272	5
45	3664	3468	369	4
46	1305	1253	100	3
47	467	454	24	3
48	172	168	8	2
49	67	66	2	2
50	26	26	0	1
51	11	11	0	1
52	5	5	0	1
53	2	2	0	1
54	1	1	0	1
55	1	1	0	1

e	#graphs	#cp	#cp1	#cp2	#with mate
0	1	1	1	-	0
1	1	1	1	-	0
2	2	2	2	-	0
3	4	4	4	-	0
4	6	5	4	1	2
5	6	6	6	-	0
6	6	6	6	-	0
7	4	4	4	-	0
8	2	2	2	-	0
9	1	1	1	-	0
10	1	1	1	-	0

e	#graphs	#cp	#cp1	#cp2	#with mate
0	1	1	1	-	0
1	1	1	1	-	0
2	2	2	2	-	0
3	5	5	5	-	0
4	9	7	5	2	4
5	15	14	13	1	2
6	21	20	19	1	2
7	24	23	22	1	2
8	24	24	24	-	0
9	21	21	21	-	0
10	15	15	15	-	0
11	9	9	9	-	0
12	5	5	5	-	0
13	2	2	2	-	0
14	1	1	1	-	0
15	1	1	1	-	0

e	#graphs	#cp	#cp1	#cp2	#cp3	#with mate
0	1	1	1	-	-	0
1	1	1	1	-	-	0
2	2	2	2	-	-	0
3	5	5	5	-	-	0
4	10	8	6	2	-	4
5	21	19	18	-	1	3
6	41	34	27	7	-	14
7	65	61	57	4	-	8
8	97	88	79	9	-	18
9	131	125	119	6	-	12
10	148	140	132	8	-	16
11	148	141	135	5	1	13
12	131	125	119	6	-	12
13	97	95	93	2	-	4
14	65	64	63	1	-	2
15	41	40	39	1	-	2
16	21	20	19	1	-	2
17	10	10	10	-	-	0
18	5	5	5	-	-	0
19	2	2	2	-	-	0
20	1	1	1	-	-	0
21	1	1	1	-	-	0

e	#graphs	#cp	#cp1	#cp2	#with mate
0	1	1	1	-	0
1	1	1	1	-	0
2	2	2	2	-	0
3	4	4	4	-	0
4	6	5	4	1	2
5	6	6	6	-	0
6	6	6	6	-	0
7	4	4	4	-	0
8	2	2	2	-	0
9	1	1	1	-	0
10	1	1	1	-	0

e	#graphs	#cp	#cp1	#cp2	#with mate
0	1	1	1	-	0
1	1	1	1	-	0
2	2	2	2	-	0
3	5	5	5	-	0
4	9	7	5	2	4
5	15	14	13	1	2
6	21	20	19	1	2
7	24	23	22	1	2
8	24	24	24	-	0
9	21	21	21	-	0
10	15	15	15	-	0
11	9	9	9	-	0
12	5	5	5	-	0
13	2	2	2	-	0
14	1	1	1	-	0
15	1	1	1	-	0

e	#graphs	#cp	#cp1	#cp2	#cp3	#with mate
0	1	1	1	-	-	0
1	1	1	1	-	-	0
2	2	2	2	-	-	0
3	5	5	5	-	-	0
4	10	8	6	2	-	4
5	21	19	18	-	1	3
6	41	34	27	7	-	14
7	65	61	57	4	-	8
8	97	88	79	9	-	18
9	131	125	119	6	-	12
10	148	140	132	8	-	16
11	148	141	135	5	1	13
12	131	125	119	6	-	12
13	97	95	93	2	-	4
14	65	64	63	1	-	2
15	41	40	39	1	-	2
16	21	20	19	1	-	2
17	10	10	10	-	-	0
18	5	5	5	-	-	0
19	2	2	2	-	-	0
20	1	1	1	-	-	0
21	1	1	1	-	-	0