Four different views of Glasgow University Tower
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
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