Four different views of Glasgow University Tower
Numbers of characteristic polynomials and cospectral graphs with respect to the Laplacian of A

 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 6 156 154 4 2 7 1044 976 130 3 8 12346 11398 1767 6 9 274668 251802 42595 9 10 12005168 11249513 1412438 23 11 1018997864 970760558 91274836 33 12 165091172592 159955412426 9854131240 226

In what follows below e denotes the number of edges, #cp the number of distinct characteristic polynomials and maxsz is the maximum number of graphs that have a cospectral mate.

## n=4

 e #graphs #cp with mate maxsz 0 1 1 0 1 1 1 1 0 1 2 2 2 0 1 3 3 3 0 1 4 2 2 0 1 5 1 1 0 1 6 1 1 0 1

## n=5

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

## n=6

 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 9 9 0 1 5 15 15 0 1 6 21 21 0 1 7 24 23 2 2 8 24 23 2 2 9 21 21 0 1 10 15 15 0 1 11 9 9 0 1 12 5 5 0 1 13 2 2 0 1 14 1 1 0 1 15 1 1 0 1

## 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 38 5 3 7 65 60 10 2 8 97 88 18 2 9 131 121 20 2 10 148 141 12 3 11 148 141 12 3 12 131 121 20 2 13 97 88 18 2 14 65 60 10 2 15 41 38 5 3 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 52 7 3 7 115 108 14 2 8 221 201 39 3 9 402 365 69 4 10 663 613 91 4 11 980 917 115 5 12 1312 1221 170 5 13 1557 1434 233 4 14 1646 1488 291 6 15 1557 1434 233 4 16 1312 1221 170 5 17 980 917 115 5 18 663 613 91 4 19 402 365 69 4 20 221 201 39 3 21 115 108 14 2 22 56 52 7 3 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 59 7 3 7 148 138 19 3 8 345 314 60 3 9 771 688 156 4 10 1637 1470 306 4 11 3252 2962 535 5 12 5995 5482 954 6 13 10120 9235 1656 6 14 15615 14191 2642 9 15 21933 19995 3597 7 16 27987 25639 4373 6 17 32403 29901 4674 6 18 34040 31564 4637 6 19 32403 29901 4674 6 20 27987 25639 4373 6 21 21933 19995 3597 7 22 15615 14191 2642 9 23 10120 9235 1656 6 24 5995 5482 954 6 25 3252 2962 535 5 26 1637 1470 306 4 27 771 688 156 4 28 345 314 60 3 29 148 138 19 3 30 63 59 7 3 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 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 62 7 3 7 165 154 21 3 8 428 389 75 3 9 1103 976 237 4 10 2769 2455 568 5 11 6759 6056 1279 5 12 15772 14304 2722 6 13 34663 31732 5455 7 14 71318 65735 10428 9 15 136433 126391 18826 10 16 241577 224880 31373 8 17 395166 369658 47972 11 18 596191 559706 68692 13 19 828728 779699 92350 12 20 1061159 997576 119163 15 21 1251389 1173357 145233 23 22 1358852 1271581 161818 19 23 1358852 1271581 161818 19 24 1251389 1173357 145233 23 25 1061159 997576 119163 15 26 828728 779699 92350 12 27 596191 559706 68692 13 28 395166 369658 47972 11 29 241577 224880 31373 8 30 136433 126391 18826 10 31 71318 65735 10428 9 32 34663 31732 5455 7 33 15772 14304 2722 6 34 6759 6056 1279 5 35 2769 2455 568 5 36 1103 976 237 4 37 428 389 75 3 38 165 154 21 3 39 66 62 7 3 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 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 67 63 7 3 7 172 161 21 3 8 467 425 79 4 9 1305 1157 276 4 10 3664 3227 793 5 11 10250 9072 2128 6 12 28259 25251 5511 6 13 75415 68300 13095 7 14 192788 177042 29242 14 15 467807 434203 62858 15 16 1069890 1000573 130173 13 17 2295898 2159096 257453 13 18 4609179 4353188 483437 13 19 8640134 8186804 858901 12 20 15108047 14342950 1450889 16 21 24630887 23403071 2325548 28 22 37433760 35577973 3508525 21 23 53037356 50428733 4926016 21 24 70065437 66682056 6390926 29 25 86318670 82268821 7659772 33 26 99187806 94676147 8546114 31 27 106321628 101581920 8985654 32 28 106321628 101581920 8985654 32 29 99187806 94676147 8546114 31 30 86318670 82268821 7659772 33 31 70065437 66682056 6390926 29 32 53037356 50428733 4926016 21 33 37433760 35577973 3508525 21 34 24630887 23403071 2325548 28 35 15108047 14342950 1450889 16 36 8640134 8186804 858901 12 37 4609179 4353188 483437 13 38 2295898 2159096 257453 13 39 1069890 1000573 130173 13 40 467807 434203 62858 15 41 192788 177042 29242 14 42 75415 68300 13095 7 43 28259 25251 5511 6 44 10250 9072 2128 6 45 3664 3227 793 5 46 1305 1157 276 4 47 467 425 79 4 48 172 161 21 3 49 67 63 7 3 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 64 7 3 7 175 164 21 3 8 485 442 81 4 9 1405 1247 295 4 10 4191 3684 914 5 11 12763 11218 2769 6 12 39243 34679 8262 7 13 119890 107348 22834 8 14 359307 326850 59682 14 15 1043774 963634 148946 15 16 2911086 2720244 357964 16 17 7739601 7300147 829099 18 18 19515361 18538338 1852312 16 19 46505609 44420144 3972194 24 20 104504341 100232767 8169019 23 21 221147351 212757285 16090529 28 22 440393606 424615658 30307025 30 23 825075506 796772409 54405586 27 24 1454265734 1406062027 92693467 31 25 2411961516 2334289130 149398201 47 26 3765262970 3646948156 227605271 45 27 5534255092 5363634223 328223775 58 28 7661345277 7427794910 449166015 67 29 9992340187 9688227140 584447282 84 30 12281841209 11905491932 722395037 92 31 14229503560 13788660723 844980527 108 32 15542350436 15056092725 930907383 138 33 16006173014 15503397759 962041146 226 34 15542350436 15056092725 930907383 138 35 14229503560 13788660723 844980527 108 36 12281841209 11905491932 722395037 92 37 9992340187 9688227140 584447282 84 38 7661345277 7427794910 449166015 67 39 5534255092 5363634223 328223775 58 40 3765262970 3646948156 227605271 45 41 2411961516 2334289130 149398201 47 42 1454265734 1406062027 92693467 31 43 825075506 796772409 54405586 27 44 440393606 424615658 30307025 30 45 221147351 212757285 16090529 28 46 104504341 100232767 8169019 23 47 46505609 44420144 3972194 24 48 19515361 18538338 1852312 16 49 7739601 7300147 829099 18 50 2911086 2720244 357964 16 51 1043774 963634 148946 15 52 359307 326850 59682 14 53 119890 107348 22834 8 54 39243 34679 8262 7 55 12763 11218 2769 6 56 4191 3684 914 5 57 1405 1247 295 4 58 485 442 81 4 59 175 164 21 3 60 68 64 7 3 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