Counting Unlabeled Chordal Graphs by Equivariant Evaporation

Abstract

We compute the number of unlabeled chordal graphs on $n$ vertices, both the total count (OEIS A048193) and the connected count (OEIS A048192), extending two sequences whose published values had remained at $n=15$. The method is a Polya-Burnside enumeration: the number of unlabeled graphs in a class closed under relabeling is the average over $S_n$ of the number of labeled graphs fixed by each permutation. The technical core is the evaluation, for an arbitrary permutation $π$, of the number of $π$-invariant labeled chordal graphs. We give a dynamic program for this quantity that lifts the evaporation-based labeled chordal counting of Hebert-Johnson, Lokshtanov and Vigoda to the equivariant setting. Its central structural ingredient is a divisor-bundle decomposition: when a connected piece spans a cyclic orbit of size $c$, it forms, for each divisor $d \mid c$, a $d$-fold bundle whose constituent is an object of the same kind in the cyclic world of order $c/d$, computed by the same program recursively. We prove the decomposition and the correctness of the resulting recurrences, and we prove that the full Burnside computation runs in sub-exponential time $n^{O(\sqrt{n})}$. We report the new terms through $n=20$ and describe four independent validations, including exact agreement with all previously known values of both sequences and an Euler-transform consistency check.