Incidence graphs constructed from t-designs

Abstract

Let D be a non-trivial simple t-design. For 1 ≤ s ≤ t, we generalize the concept of the incidence graph of D and construct a new bipartite regular graph Ζ. We obtain that the edge-transitivity of the graph Ζ is equivalent to the s-flag-transitivity of the design D. We then, for s = 2, classify the semisymmetric graphs among the graphs Ζ constructed from biplanes and triplanes. Finally, we study the connectedness and the energy of incidence graphs. Several open problems are proposed, one of which asks whether the incidence graphs have large vertex-connectivity.