Polynomial Characterizations of Distance‐Biregular Graphs

ABSTRACT Fiol, Garriga, and Yebra introduced the notion of pseudo‐distance‐regular vertices, which they used to come up with a new characterization of distance‐regular graphs. Building on that work, Fiol and Garriga developed the spectral excess theorem for distance‐regular graphs. We extend both these characterizations to distance‐biregular graphs and show how these characterizations can be used to study bipartite graphs with distance‐regular halved graphs and graphs with the spectrum of a distance‐biregular graph.