A general construction of strictly Neumaier graphs and a related switching

TL;DR

This paper introduces a new general method for constructing strictly Neumaier graphs with nexus 1, unifies previous constructions, and demonstrates how to generate cospectral graphs and small examples with geometric or algebraic descriptions.

Contribution

It provides a unified construction method for Neumaier graphs, extends known constructions, and explores cospectral graphs and small examples with detailed descriptions.

Findings

Constructed Neumaier graphs with nexus 1 using a generalized approach.

Showed how to produce cospectral Neumaier graphs via switching.

Derived small strictly Neumaier graphs with geometric or algebraic characterizations.

Abstract

We present a construction of Neumaier graphs with nexus 1, which generalises two known constructions of Neumaier graphs. We also use W. Wang, L. Qiu, and Y. Hu switching to show that we construct cospectral Neumaier graphs. Finally, we show that several small strictly Neumaier graphs can be obtained from our construction, and give a geometric or algebraic description for each of these graphs.