Four different views of Glasgow University Tower
Numbers of characteristic polynomials and cospectral graphs with respect to the signless 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 10 2 2 5 34 32 4 2 6 156 148 16 2 7 1044 992 102 3 8 12346 11718 1201 4 9 274668 264807 19001 5 10 12005168 11674848 636607 9 11 1018997864 999003763 38966931 20 12 165091172592 162807642839 4476641637 42

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 1 1 1 0 1 2 2 2 0 1 3 3 2 2 2 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 3 2 2 4 6 6 0 1 5 6 6 0 1 6 6 6 0 1 7 4 3 2 2 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 4 2 2 4 9 8 2 2 5 15 15 0 1 6 21 21 0 1 7 24 22 4 2 8 24 22 4 2 9 21 21 0 1 10 15 15 0 1 11 9 8 2 2 12 5 4 2 2 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 10 8 4 2 4 10 9 2 2 5 21 20 2 2 6 41 40 2 2 7 65 62 6 2 8 97 91 12 2 9 131 124 14 2 10 148 142 11 2 11 148 142 11 3 12 131 124 14 2 13 97 91 12 2 14 65 62 6 2 15 41 40 2 2 16 21 20 2 2 17 10 9 2 2 18 5 4 2 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 10 8 4 2 4 11 10 2 2 5 24 22 4 2 6 56 52 7 2 7 115 107 16 2 8 221 203 36 2 9 402 372 58 3 10 663 621 79 3 11 980 932 93 3 12 1312 1260 100 4 13 1557 1482 142 4 14 1646 1558 167 4 15 1557 1482 141 4 16 1312 1261 98 4 17 980 935 86 4 18 663 628 66 3 19 402 376 51 3 20 221 208 26 2 21 115 108 14 2 22 56 53 5 3 23 24 22 4 2 24 11 10 2 2 25 5 4 2 2 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 4 2 2 4 11 10 2 2 5 25 23 4 2 6 63 58 9 3 7 148 136 24 2 8 345 320 50 2 9 771 723 93 3 10 1637 1540 179 5 11 3252 3099 293 4 12 5995 5769 433 4 13 10120 9762 688 4 14 15615 15043 1109 4 15 21933 21093 1635 4 16 27987 26978 1958 5 17 32403 31312 2086 4 18 34040 32946 2071 4 19 32403 31319 2075 4 20 27987 26989 1940 5 21 21933 21103 1619 4 22 15615 15058 1081 4 23 10120 9779 656 4 24 5995 5783 408 4 25 3252 3107 278 4 26 1637 1558 149 4 27 771 728 84 3 28 345 326 38 2 29 148 137 22 2 30 63 59 7 3 31 25 23 4 2 32 11 10 2 2 33 5 4 2 2 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 4 2 2 4 11 10 2 2 5 26 24 4 2 6 66 60 11 3 7 165 149 31 2 8 428 388 80 2 9 1103 1022 155 4 10 2769 2587 338 6 11 6759 6399 681 4 12 15772 15090 1307 4 13 34663 33445 2344 5 14 71318 69067 4362 4 15 136433 132293 8069 5 16 241577 234458 13909 7 17 395166 383937 21814 5 18 596191 579866 31495 6 19 828728 806636 42534 7 20 1061159 1032890 54427 9 21 1251389 1217370 65430 9 22 1358852 1321256 72165 9 23 1358852 1321255 72181 9 24 1251389 1217418 65338 9 25 1061159 1032965 54290 9 26 828728 806738 42342 7 27 596191 579961 31312 6 28 395166 384023 21660 5 29 241577 234561 13716 5 30 136433 132379 7919 5 31 71318 69153 4202 4 32 34663 33510 2224 5 33 15772 15157 1182 4 34 6759 6450 590 4 35 2769 2633 259 4 36 1103 1038 128 3 37 428 398 60 2 38 165 151 27 3 39 66 61 9 3 40 26 24 4 2 41 11 10 2 2 42 5 4 2 2 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 4 2 2 4 11 10 2 2 5 26 24 4 2 6 67 61 11 2 7 172 155 33 3 8 467 420 93 3 9 1305 1198 205 4 10 3664 3405 487 6 11 10250 9671 1092 5 12 28259 26924 2536 4 13 75415 72567 5436 6 14 192788 186703 11713 6 15 467807 454829 25201 6 16 1069890 1042608 53338 10 17 2295898 2240997 107427 7 18 4609179 4505188 203086 8 19 8640134 8455252 360278 10 20 15108047 14798366 602866 11 21 24630887 24139559 956313 12 22 37433760 36695764 1436639 16 23 53037356 51993405 2033387 12 24 70065437 68687523 2686072 12 25 86318670 84631681 3290167 13 26 99187806 97269707 3739927 20 27 106321628 104282252 3973874 20 28 106321628 104282351 3973697 20 29 99187806 97269919 3739550 20 30 86318670 84632027 3289553* 13 31 70065437 68687924 2685316 12 32 53037356 51993928 2032407 12 33 37433760 36696381 1435463 16 34 24630887 24140330 954838 12 35 15108047 14799173 601329 9 36 8640134 8456061 358747 10 37 4609179 4505871 201801 8 38 2295898 2241532 106447 7 39 1069890 1043007 52593 10 40 467807 455109 24695 6 41 192788 186936 11291 6 42 75415 72727 5152 6 43 28259 27074 2261 4 44 10250 9759 936 4 45 3664 3467 378 4 46 1305 1217 173 3 47 467 431 71 2 48 172 157 29 3 49 67 62 9 3 50 26 24 4 2 51 11 10 2 2 52 5 5 0 1 53 2 2 0 1 54 1 1 0 1 55 1 1 0 1

The * indicates a correction to the figure of 3289555 that was given in the paper by Haemers and Spence.

## 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 4 2 2 4 11 10 2 2 5 26 24 4 2 6 68 62 11 3 7 175 157 35 3 8 485 434 100 3 9 1405 1276 246 4 10 4191 3854 630 6 11 12763 11934 1547 7 12 39243 37066 4094 6 13 119890 114618 9991 8 14 359307 346329 24811 6 15 1043774 1012573 60189 8 16 2911086 2836427 145203 10 17 7739601 7564465 342501 8 18 19515361 19117238 780313 8 19 46505609 45636700 1703279 10 20 104504341 102693370 3548703 11 21 221147351 217548897 7051462 12 22 440393606 433580546 13355795 16 23 825075506 812775813 24125153 22 24 1454265734 1433132369 41477954 22 25 2411961516 2377512373 67645822 20 26 3765262970 3712180245 104255411 20 27 5534255092 5457060270 151584296 20 28 7661345277 7555397762 207956531 24 29 9992340187 9854950806 269523097 40 30 12281841209 12113414975 330197523 36 31 14229503560 14034414540 382205312 36 32 15542350436 15329117432 417514154 41 33 16006173014 15786494843 430036552 42 34 15542350436 15329126504 417497214 41 35 14229503560 14034428220 382179425 36 36 12281841209 12113434248 330160980 36 37 9992340187 9854975519 269476213 40 38 7661345277 7555427319 207900721 24 39 5534255092 5457086052 151535437 20 40 3765262970 3712201659 104214784 20 41 2411961516 2377528881 67614383 20 42 1454265734 1433146706 41450407 22 43 825075506 812786998 24103521 22 44 440393606 433590429 13336634 16 45 221147351 217557006 7035791 12 46 104504341 102699904 3536170 9 47 46505609 45640913 1695271 10 48 19515361 19120093 774972 8 49 7739601 7566265 339177 8 50 2911086 2837794 142695 10 51 1043774 1013403 58709 8 52 359307 347034 23545 6 53 119890 115034 9269 6 54 39243 37415 3477 6 55 12763 12082 1300 4 56 4191 3935 490 4 57 1405 1300 205 4 58 485 446 76 3 59 175 159 31 3 60 68 63 9 3 61 26 24 4 2 62 11 10 2 2 63 5 4 2 2 64 2 2 0 1 65 1 1 0 1 66 1 1 0 1