$q$-Analogs of divisible design graphs and Deza graphs

TL;DR

This paper introduces $q$-analogs of divisible design graphs and Deza graphs, establishing their origins from spreads and characterizing their parameters and examples, thus extending the theory of these combinatorial structures.

Contribution

It defines $q$-analogs of divisible design graphs and Deza graphs, linking them to spreads and providing classifications and examples for the first time.

Findings

All $q$-analogs of divisible design graphs derive from spreads.

Characterization of non-strongly regular $q$-analogs of Deza graphs with minimal parameters.

Examples of $q$-analogs of Deza graphs and their parameter sets.

Abstract

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.