q-Analogs of divisible design graphs and Deza graphs

Divisible design graphs were introduced in 2011 by Haemers, Kharaghani and Meulenberg. In this paper, we introduce the notion of q-analogs of divisible design graphs and show that all q-analogs of divisible design graphs come from spreads, and are actually q-analogs of strongly regular graphs. Deza graphs were introduced by Erickson, Fernando, Haemers and Hardy in 1999. In this paper, we introduce q-analogs of Deza graphs. Further, we determine possible parameters, give examples of q-analogs of Deza graphs and characterize all non-strongly regular q-analogs of Deza graphs with the smallest parameters.