Cospectral data

Four different views of Glasgow University Tower

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Graphs on <= 12
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Cospectral Data Data wrt Laplace
Spectrum Data wrt the
signless Laplacian Data wrt both A
and its complement

Andries Brouwer's
data

Numbers of cospectral graphs and characteristic polynomials of graphs G for which both G and its complement have cospectral mates

n	#graphs	#char.pols.	#graphs with mate	maxsz
0	1	1	0	1
1	1	1	0	1
2	2	2	0	1
3	4	4	0	1
4	11	11	0	1
5	34	34	0	1
6	156	156	0	1
7	1044	1024	40	2
8	12346	11750	1166	4
9	274668	251402	43811	10
10	12005168	10691782	2418152	21
11	1018997864	902890368	212264372	46
12	165091172592	148257594204	30955767527	128

In what follows below e denotes the number of edges, #cp the number of distinct pairs of characteristic polynomials corresponding to G and its complement, #with mate the number of graphs G such that both G and its complement have cospectral mates, and maxsz is the maximum number of graphs G for which G and its complement have the same pair of characteristic polynomials (P,Q) say.

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

n < 7
No cospectral graphs

n = 7

#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	10	10	0	1
5	21	21	0	1
6	41	40	2	2
7	65	64	2	2
8	97	95	4	2
9	131	128	6	2
10	148	145	6	2
11	148	145	6	2
12	131	128	6	2
13	97	95	4	2
14	65	64	2	2
15	41	40	2	2
16	21	21	0	1
17	10	10	0	1
18	5	5	0	1
19	2	2	0	1
20	1	1	0	1
21	1	1	0	1

n = 8

#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	11	0	1
5	24	24	0	1
6	56	55	2	2
7	115	113	4	2
8	221	215	10	3
9	402	387	30	2
10	663	635	55	3
11	980	935	87	4
12	1312	1243	133	3
13	1557	1470	173	3
14	1646	1556	178	3
15	1557	1470	173	3
16	1312	1243	133	3
17	980	935	87	4
18	663	635	55	3
19	402	387	30	2
20	221	215	10	3
21	115	113	4	2
22	56	55	2	1
23	24	24	0	1
24	11	11	0	1
25	5	5	0	1
26	2	2	0	1
27	1	1	0	1
28	1	1	0	1

n = 9

#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	11	0	1
5	25	25	0	1
6	63	62	2	2
7	148	145	6	2
8	345	335	18	3
9	771	740	61	3
10	1637	1552	158	3
11	3252	3056	383	5
12	5995	5578	764	5
13	10120	9350	1469	5
14	15615	14351	2342	6
15	21933	20064	3557	6
16	27987	25492	4624	10
17	32403	29462	5619	6
18	34040	30938	5805	6
19	32403	29462	5619	6
20	27987	25492	4624	10
21	21933	20064	3557	6
22	15615	14351	2342	6
23	10120	9350	1469	5
24	5995	5578	764	5
25	3252	3056	383	5
26	1637	1552	158	3
27	771	740	61	3
28	345	335	18	3
29	148	145	6	2
30	63	62	2	2
31	25	25	0	1
32	11	11	0	1
33	5	5	0	1
34	2	2	0	1
35	1	1	0	1
36	1	1	0	1

n = 10

#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	11	0	1
5	26	26	0	1
6	66	65	2	2
7	165	162	6	2
8	428	416	22	3
9	1103	1058	86	3
10	2769	2621	278	4
11	6759	6327	831	5
12	15772	14596	2178	5
13	34663	31805	5380	6
14	71318	64904	11811	8
15	136433	123454	24094	8
16	241577	217361	44229	15
17	395166	354164	75358	12
18	596191	532285	116870	13
19	828728	738345	166403	12
20	1061159	942745	217639	21
21	1251389	1110147	260561	16
22	1358852	1205390	283328	15
23	1358852	1205390	283328	15
24	1251389	1110147	260561	16
25	1061159	942745	217639	21
26	828728	738345	166403	12
27	596191	532285	116870	13
28	395166	354164	75358	12
29	241577	217361	44229	15
30	136433	123454	24094	8
31	71318	64904	11811	8
32	34663	31805	5380	6
33	15772	14596	2178	5
34	6759	6327	831	5
35	2769	2621	278	4
36	1103	1058	86	3
37	428	416	22	3
38	165	162	6	2
39	66	65	2	1
40	26	26	0	1
41	11	11	0	1
42	5	5	0	1
43	2	2	0	1
44	1	1	0	1
45	1	1	0	1

n = 11

#e	#graphs	#cp	#with mate	maxsz
1	1	1	0	1
1	1	1	0	1
2	2	2	0	1
3	5	5	0	1
4	11	11	0	1
5	26	26	0	1
6	67	66	2	2
7	172	169	6	2
8	467	454	24	3
9	1305	1253	100	3
10	3664	3469	367	4
11	10250	9585	1266	5
12	28259	26097	4032	5
13	75415	68969	12057	6
14	192788	174857	33149	9
15	467807	421658	85356	9
16	1069890	959623	202525	15
17	2295898	2052697	446061	16
18	4609179	4109854	912308	18
19	8640134	7691824	1731073	20
20	15108047	13429388	3059756	30
21	24630887	21868085	5034203	30
22	37433760	33207505	7703505	33
23	53037356	47022524	10974450	34
24	70065437	62074160	14598628	36
25	86318670	76420613	18101952	31
26	99187806	87791373	20865502	46
27	106321628	94110915	22365864	34
28	106321628	94110915	22365864	34
29	99187806	87791373	20865502	46
30	86318670	76420613	18101952	31
31	70065437	62074160	14598628	36
32	53037356	47022524	10974450	34
33	37433760	33207505	7703505	33
34	24630887	21868085	5034203	30
35	15108047	13429388	3059756	30
36	8640134	7691824	1731073	20
37	4609179	4109854	912308	18
38	2295898	2052697	446061	16
39	1069890	959623	202525	15
40	467807	421658	85356	9
41	192788	174857	33149	9
42	75415	68969	12057	6
43	28259	26097	4032	5
44	10250	9585	1266	5
45	3664	3469	367	4
46	1305	1253	100	3
47	467	454	24	3
48	172	169	6	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

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	11	0	1
5	26	26	0	1
6	68	67	2	2
7	175	172	6	2
8	485	472	24	3
9	1405	1351	104	3
10	4191	3974	407	4
11	12763	11947	1551	5
12	39243	36256	5584	6
13	119890	109577	19273	8
14	359307	325473	62718	10
15	1043774	939453	192847	10
16	2911086	2608356	556961	15
17	7739601	6918409	1507986	17
18	19515361	17422757	3834932	23
19	46505609	41512474	9145360	26
20	104504341	93319034	20477714	37
21	221147351	197626664	43058050	42
22	440393606	393893201	85120983	52
23	825075506	738676958	158209341	75
24	1454265734	1303120069	276858615	84
25	2411961516	2162795213	456628599	70
26	3765262970	3378196960	709763445	88
27	5534255092	4967908688	1039364150	77
28	7661345277	6879928549	1435193209	98
29	9992340187	8975124938	1869987036	87
30	12281841209	11032010470	2299289628	89
31	14229503560	12782278829	2664184869	106
32	15542350436	13963409883	2907846283	98
33	16006173014	14381233724	2993148173	128
34	15542350436	13963409883	2907846283	98
35	14229503560	12782278829	2664184869	106
36	12281841209	11032010470	2299289628	89
37	9992340187	8975124938	1869987036	87
38	7661345277	6879928549	1435193209	98
39	5534255092	4967908688	1039364150	77
40	3765262970	3378196960	709763445	88
41	2411961516	2162795213	456628599	70
42	1454265734	1303120069	276858615	84
43	825075506	738676958	158209341	75
44	440393606	393893201	85120983	52
45	221147351	197626664	43058050	42
46	104504341	93319034	20477714	37
47	46505609	41512474	9145360	26
48	19515361	17422757	3834932	23
49	7739601	6918409	1507986	17
50	2911086	2608356	556961	15
51	1043774	939453	192847	10
52	359307	325473	62718	10
53	119890	109577	19273	8
54	39243	36256	5584	6
55	12763	11947	1551	5
56	4191	3974	407	4
57	1405	1351	104	3
58	485	472	24	3
59	175	172	6	2
60	68	67	2	1
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