Equal Lagrange Numbers with Non-Isomorphic Band Graphs: A Counterexample to Schiffler's Problem 6.3

Abstract

We consider the following question: if two lattice paths in the same set $\mathcal{D}(a,b)$ have the same Lagrange number, must their band graphs be isomorphic? We exhibit two explicit lattice paths in $\mathcal{D}(17,9)$ with the same Lagrange number but non-isomorphic associated band graphs, thereby giving a counterexample to this question.