Modulo Errors

As a companion to my previous post, here’s the list of valid forms of a connected maximal cyclotomic graph with an entry from the ring of integers of Q(√ -7):

Uncharged Lines:

Uncharged Squares:

Uncharged Hexagons:

Uncharged Cubes:

T_2k Variants (Infinite Family):

A chain of the form

for any integer k.

Charged Triangles:

Charged Squares:

or

C_2k Variants (Infinite Family):

A chain of the form

for any integer k.

To recap: I’ve been trying to completely classify the possible matrices/graphs subject to a constraint on their eigenvalues we’re describing as cyclotomicity. This is a problem that can be posed in the ring of integers of any imaginary quadratic extension field, but for all but finitely many of them reduces to the problem in the rational-integer case which has been solved in a paper by my supervisor.

For a couple of the remaining fields, the problem is easy: there’s only a finite supply of graphs featuring a non-rational integer label, which can be found simply by running a growing process to termination. But once you move to fields with norm 2 integers, there’s enough freedom for things to get interesting: I’ve been working in the rings of integers of Q(√ -2) and Q(√ -7), where I can demonstrate an infinite family of such graphs and so the growing algorithm can never terminate. Nonetheless, in the simpler, ‘uncharged’ version of the problem, I have (proven) a complete classification in both of these fields. With a bit more work I’ve now settled the general case in Q(√ -2) and expect the logic of the argument (although not the precise results) to carry over to Q(√ -7).

That argument is essentially a lengthy case analysis; rather than get into the details I thought I’d just present the ‘zoo’ of possible graphs. The forms presented are necessary conditions (any cyclotomic graph will take one of these forms) but not sufficient (there may be non-cyclotomic graphs satisfying the form). However, no form contains no cyclotomic graphs - for a given form, including any instance of the infinite ones, I can exhibit at least one class of cyclotomic graphs.

After applying a numbering, the visual styling of an edge between two nodes i<j indicates the norm of the edge label (entry [i,j] of the matrix; take the conjugate for entry [j,i]); uncharged nodes are indicated by a point whilst [C] denotes a charged node (value of ±1 for entry [i.i] if node i is charged, otherwise zero). The precise choice of labels and charges requires some care over signs to ensure that the matrix has minimal polynomial x²-4; working with forms saves tracking such details, choosing equivalence class representatives, etc.

Then for the ring of integers of Q(√ -2) any connected maximal cyclotomic graph with a non-rational integer label must take one of the following forms:

Uncharged Squares:

Uncharged Cubes:

T_2k Variants (Infinite Family):

A chain of the form

for any integer k.

Charged Triangles:

Charged Squares:

or

C_2k Variants (Infinite Family):

A chain of the form

for any integer k.

As I said, I expect this to easily generalise to Q(√-7); the remaining fields Q(√-1) and Q(√-3) present more of a computational challenge, but meeting that challenge will hopefully be rewarded with more interesting behaviour!