Regular Graphs with Pairwise Distinct K_t-Degrees

Abstract

For a fixed integer \(t \ge 3\), let \(\kappa_t(v)\) denote the number of copies of \(K_t\) containing a vertex \(v\). We prove that for every \(t \ge 6\) and every sufficiently large even integer \(r\), there exists an explicit connected \(r\)-regular graph on \(2r+2\) vertices whose vertex \(K_t\)-degrees are pairwise distinct. The construction is uniform in \(t\) and \(r\) and has order linear in the regularity. Its analysis uses exact binomial clique-count formulas, a reflection order that localises possible collisions, polynomial noncancellation, and a uniform root bound. For \(6 \le t \le 15\), exact integer certification of the same construction improves the sufficient condition to every even \(r \ge 12(t-2)\). This record contains the manuscript together with the mathematical supplementary material and the exact-arithmetic verification archive supporting the fixed-order result. The verification archive includes certificate streams, reconstruction code, integrity metadata, and exact finite checks.