Mohib
Ahmad

Preprint · not yet submitted

Weighted certificates over real and finite fields for generalized Petersen graphs

There is a graph invariant that is hard to pin down, and a conjecture that it equals 8 for a whole infinite family of graphs. Working over the real numbers, this paper proves the conjecture outright for a sixth of all cases, by building matrices whose singularity is decided by where eight roots of a polynomial land on the unit circle — and then proves that this technique cannot finish the job, because an arithmetic obstruction rules out the rest. The revision changes the number system. Over finite fields the same construction reaches many more cases, raising the proved share to 46%, but it buys that reach at a price the paper states rather than hides: on the new families it settles the zero forcing number and not the maximum nullity. Establishing the limits of your own method, twice, is the less common half.

The shape rotating behind this section is P(n,3) — the graph the paper is about. Its outer cycle, spokes, and step-3 inner chords are drawn exactly as defined, and n cycles through 10, 24, 28, 42: the four divisibility families for which the certificates prove M(P(n,3)) = Z(P(n,3)) = 8. The chords change color with n, and every color on this site means one of these four.