7
$\begingroup$

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?
$\endgroup$
3
  • $\begingroup$ Along $b + c =4a$, $\Delta$ is a square times $b^2 + 18 bc + c^2$, so there is a solvable conic (isomorphic to $b^2 + 18bc + c^2 = z^2$) above the line $b + c = 4a$ in the surface $\Delta(a,b,c) = z^2$. This conic may well account for the majority of the rational points on your surface. $\endgroup$ – Michael Stoll Dec 7 '15 at 12:11
  • $\begingroup$ If $p$ is an odd prime divisor of $a$ and $b+c = 4a$, then $c \equiv -b \bmod p$, hence $p \mid (4b)^2 + z^2$, so $p \equiv 1 \bmod 4$. $\endgroup$ – Michael Stoll Dec 7 '15 at 12:14
  • $\begingroup$ @MichaelStoll You are right, so in fact the "regular" solutions are easy, as they can be cast as solutions of $(b+c)^2+16bc=z^2$. But isn't it puzzling that there are those "sporadic" solutions beside them? $\endgroup$ – Wolfgang Dec 7 '15 at 12:40
10
$\begingroup$

I take this from my comments above and add something.

The question is about rational points on the surface $S$ given by $$ \Delta(a,b,c) := (ab+bc+ca)^3 - 27(abc)^2 = z^2 $$ in the weighted projective space ${\mathbb P}_{1,1,1,3}$. Along the line $b+c = 4a$ the left hand side factors as a square times $b^2 + 18bc + c^2$, and the equation effectively reduces to $b^2 + 18bc + c^2 = z^2$, which is a solvable conic and has lots of rational points that can be parameterized. Also, for these points, any odd prime divisor of $a$ satisfies $p \mid (4b)^2 + z^2$, which implies (for coprime $a,b,c$) that $p \equiv 1 \bmod 4$.

There are other curves on $S$. For example, considering lines through the point $(1:1:1)$ (which is a simple double point of (the rational curve) $\Delta = 0$), we obtain curves of the form $z^2 = q(x,y)$, where $q$ is a quartic. If we take the line $4a = b + 3c$, this results in a curve birational to $$ y^2 = 64 x^4 + 480 x^3 + 912 x^2 + 640 x + 117, $$ which is a curve of genus 1 with some rational points. Doing the math, we find that this curve is isomorphic to the elliptic curve $y^2 + y = x^3 - 1$ (243a1 in the Cremona database) of Mordell-Weil rank 1. This implies that there are infinitely many rational points on this curve. The smallest points with strictly positive coordinates coming from this curve have (after re-ordering) $$(a,b,c) = (2,3,6), \quad (6,57,74), \quad (2068,7657,9520).$$

More generally, considering the pencil of lines through $(1:1:1)$ gives us a fibration of $S$ into genus 1 curves that admits a horiziontal curve with a degree 2 map to the base; this curve is the conic from above. So after a base-change of degree 2, we actually have a fibration into elliptic curves, and one would expect at least half of them to have positive rank, resulting in infinitely many genus 1 curves on $S$ with infinitely many rational points on each. For example, your first starred solution $(5, 12, 77)$ is on a curve isomorphic to $y^2 = x^3 - 41659288807500$, which has even rank 2.

$\endgroup$
1
  • 1
    $\begingroup$ Thank you! For 4a=b+3c, the 3rd point (2068,7657,9520) even yields integer eigenvalues. :) $\endgroup$ – Wolfgang Dec 7 '15 at 14:08

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.