This talk is based on to be published joint work with Aida Abiad and Bart De Bruyn.

Hoffman’s bound is a well-known eigenvalue bound on the chromatic number of a graph. Graphs attaining this bound (Hoffman colorable graphs) are known to be very structured. In this talk, we shift perspective and view the Hoffman bound as a graph parameter instead of as a spectral bound; and call it Hoffman number instead of Hoffman bound accordingly. We investigate Hoffman colorability and the Hoffman number for several notions of regularity in graphs: regular, (co-)edge-regular, and strongly regular.

This shift of perspective has various interesting consequences. For one, it allows to relate Hoffman colorability in strongly regular graphs to the notions of pseudo-geometricity and unique vector colorability, and we classify the strongly regular graphs with Hoffman number at most 3. Furthermore, using Hoffman numbers we give a new characterization of strongly regular graphs among regular graphs. Lastly, in the case of co-edge-regular graphs, we give an improvement of Hoffman’s bound.