This is inspired by a recent question about complete multipartite integral graphs. I am wondering if more can be said about tripartite integral graphs with block sizes $a<b<c$. It is easy to see that their three nonzero eigenvalues are the same as those of $$\begin{pmatrix} 0&b&c\\ a&0&c\\ a&b&0 \end{pmatrix}$$ because the corresponding eigenvectors must have the same entry inside each block. From Cardano's formula we get as a necessary condition for integer eigenvalues of this $3\times3$ matrix that $$\Delta=(ab+ac+bc)^3-27(abc)^2$$must be a square.

(Or, in terms of their reciprocals $u,v,w$, which can be wlog required to be integers, $$\Delta'=\frac{(u+v+w)^3}{uvw}-27$$ must be a square.)

Note that this condition is not sufficient; e.g. for $c<550$ it is fulfilled by $86$ coprime triples, but only for $71$ of them the eigenvalues are integers.

Here are the triples for $c<550$.

```
a,b c λ1 λ2 λ3
5 8 12 -10 -6 16
* 5 12 77 -33 -7 40
13 4 48 -26 -6 32
13 7 45 -26 -9 35
13 24 28 -26 -16 42
17 3 65 -34 -5 39
17 12 56 -34 -14 48
17 33 35 -34 -21 55
25 9 91 -50 -13 63
25 12 88 -50 -16 66
29 36 80 -58 -32 90
29 39 77 -58 -33 91
37 15 133 -74 -21 95
37 16 132 -74 -22 96
37 63 85 -74 -45 119
41 11 153 -82 -17 99
41 24 140 -82 -30 112
41 60 104 -82 -48 130
* 45 136 396 -240 -66 306
* 49 220 441 -315 -77 392
53 5 207 -106 -9 115
53 44 168 -106 -48 154
53 95 117 -106 -65 171
61 13 231 -122 -21 143
61 40 204 -122 -48 170
61 84 160 -122 -70 192
65 8 252 -130 -14 144
65 51 209 -130 -57 187
65 99 161 -130 -77 207
73 84 208 -146 -78 224
73 105 187 -146 -85 231
85 21 319 -170 -33 203
85 112 228 -170 -96 266
89 20 336 -178 -32 210
89 57 299 -178 -69 247
89 176 180 -178 -110 288
97 115 273 -194 -105 299
97 136 252 -194 -112 306
101 63 341 -202 -77 279
101 143 261 -202 -117 319
101 24 380 -202 -38 240
109 96 340 -218 -102 320
109 189 247 -218 -133 351
109 7 429 -218 -13 231
113 144 308 -226 -126 352
113 84 368 -226 -96 322
113 17 435 -226 -29 255
125 221 279 -250 -153 403
* 125 357 500 -425 -175 600
137 203 345 -274 -161 435
137 152 396 -274 -144 418
145 112 468 -290 -126 416
145 99 481 -290 -117 407
149 260 336 -298 -182 480
149 69 527 -298 -93 391
149 56 540 -298 -80 378
157 276 352 -314 -192 506
157 135 493 -314 -145 459
169 220 456 -338 -190 528
169 217 459 -338 -189 527
173 285 407 -346 -209 555
181 220 504 -362 -198 560
185 308 432 -370 -224 594
193 364 408 -386 -238 624
197 255 533 -394 -221 615
* * 200 44 525 -330 -70 400
205 391 429 -410 -253 663
* 225 200 252 -240 -210 450
233 464 468 -466 -288 754
241 451 513 -482 -297 779
```

Note that most often, $\lambda_1=-2a$ or $\lambda_1=-2b$ (BTW the table is sorted by this $a$ or $b$ in the first column, sometimes swapping $a$ and $b$). If we denote the entries of the first column by $a$, all these values have only divisors of the form $4k+1$, moreover each time $b+c=4a$. Further, $\lambda_2$ (and also $\lambda_3$, because $\lambda_1$ is even and the trace is $0$) has the same parity as $b$. I suppose all this can be explained easily by somebody who is knowledgeable about the properties of the related elliptical curve.

**But the really intriguing thing are the exceptions where $-\lambda_1$ is not the double of $a$ or $b$. They are marked by $*$.**

What about these exceptional triples? Are there infinitely many of them?

For the regular triples, does each prime $p=4k+1$ occur at least three times as one of $a$ or $b$?

(So far, $13$ and $17$ even occur four times)Are there infinite families of co-prime triples which can be explicitely given (e.g. by a recursion)?

I have also done a similar exhaustive search for 4-partite graphs with block sizes $a<b<c<d$. For $d<80$, there are none, but there are some, e.g. $K_{8,15,32,40\ \ }$ or $K_{3,35,48,77\ \ }$, which have two integer nonzero eigenvalues.

- Do such 4-partite integral graphs exist?