Hoffman's coclique bound for normal regular digraphs, and nonsymmetric association schemes

TL;DR

This paper extends Hoffman's coclique bound to normal regular digraphs, characterizes certain Hadamard matrices via digraphs, and explores decompositions into cocliques within nonsymmetric association schemes.

Contribution

It generalizes Hoffman's bound to a broader class of digraphs and links these structures to nonsymmetric association schemes and Hadamard matrices.

Findings

Characterization of skew-Bush-type Hadamard matrices via digraphs

Identification of normal digraphs with cocliques attaining the bound

Decomposition of vertex sets into disjoint cocliques in relation graphs

Abstract

We extend Hoffman's coclique bound for regular digraphs with the property that its adjacency matrix is normal, and discuss cocliques attaining the inequality. As a consequence, we characterize skew-Bush-type Hadamard matrices in terms of digraphs. We present some normal digraphs whose vertex set is decomposed into disjoint cocliques attaining the bound. The digraphs provided here are relation graphs of some nonsymmetric association schemes.