Divisible design graphs with selfloops

TL;DR

This paper develops a foundational theory for divisible design graphs with selfloops, introduces infinite families, and provides methods for constructing new examples, advancing understanding of their spectral and structural properties.

Contribution

It introduces the concept of divisible design graphs with selfloops, describes infinite families, and presents dual Seidel switching for constructing new graphs.

Findings

Classified all examples with certain parameters or up to three eigenvalues

Analyzed the spectrum and structure of these graphs

Developed a procedure for constructing new graphs from existing ones

Abstract

We develop a basic theory for divisible design graphs with possible selfloops (LDDG's), and describe two infinite families of such graphs, some members of which are also classical examples of divisible design graphs without loops (DDG's). Among the described theoretical results is a discussion of the spectrum, a classification of all examples satisfying certain parameter restrictions or having at most three eigenvalues, a discussion of the structure of the improper and the disconnected examples, and a procedure called dual Seidel switching which allows to construct new examples of LDDG's from others.