Parameters for certain locally-regular graphs

TL;DR

This paper investigates the existence conditions for $(k,t)$-regular graphs, which are graphs where each vertex has degree $k$ and the neighborhood of each vertex forms a $t$-regular subgraph, providing new theoretical insights and examples.

Contribution

The paper establishes new existence conditions for $(k,t)$-regular graphs and supplies a broad set of examples illustrating these conditions.

Findings

Derived new existence criteria for $(k,t)$-regular graphs.

Provided numerous examples of such graphs.

Enhanced understanding of the structure of locally-regular graphs.

Abstract

A graph is called $(k,t)$-regular if it is $k$-regular and the induced subgraph on the neighbourhood of every vertex is $t$-regular. We find new conditions on $(k,t)$ for the existence of such graphs and provide a wide range of examples.