Patterns Admitting Regular Graphs with Pairwise Distinct F Degrees

For a fixed graph F, the F degree of a vertex in a host graph G is the number of subgraphs of G isomorphic to F that contain that vertex. This work classifies the connected patterns that admit a regular host in which all vertex F degrees are pairwise distinct. For every finite simple connected graph F with at least three vertices, such a host exists if and only if F is not a star. Moreover, every connected nonstar F admits infinitely many connected regular hosts with pairwise distinct F degrees, and these hosts may all be chosen with odd regularity and diameter two. The proof combines lexicographic product constructions with complement expansions for rooted subgraph counts, partition lattice inversion, and a leading scale analysis. The remaining short girth cases are handled using finite exact separation certificates together with an irrational rotation argument. The accompanying supplementary files contain the explicit seed graphs and the verification code for the finite computations used in the proof.