The Distance Spectrum Does Not Determine Bipartiteness

Over a decade ago, Koolen, Hayat, and Iqbal posed the problem of whether the distance spectrum determines bipartiteness within the class of connected graphs. In this paper, we resolve this problem in the negative: we explicitly construct an infinite family of counterexamples, where each pair comprises a connected bipartite graph and a connected non-bipartite graph with equal distance spectra.