Divisible design graphs with selfloops

TL;DR

This work extends divisible design graphs to allow loops (LDDG), establishing a coherent theory with strong parameter relations and spectral structure. It introduces two infinite, geometry-based families of LDDG built from polarities in finite projective spaces, and analyzes their canonical partitions, loop counts, and complements. The spectrum is shown to contain at most five eigenvalues, with explicit relations among eigenvalues via $A^2=(k-\lambda_1)I_v+\lambda_2 J_v+(\lambda_1-\lambda_2)K_{(m,n)}$, plus a constructive framework using a $2$-order automorphism (dual Seidel switching) to generate new examples. Together with classification results for cases like $\lambda_1=k$ or at most three eigenvalues, the paper provides a practical toolkit for creating and transforming LDDG's from projective-geometric data, and connects these graphs to known DDG and Deza-type graphs through polarity-based constructions.

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.

Theorems & Definitions (77)

• Lemma 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 3.1
• proof
• Proposition 3.2
• Proposition 3.3
• proof
• Proposition 3.4
• proof