On a composition of digraphs

TL;DR

This paper introduces the diagonal union composition of digraphs, a method to construct digraphs of unitary matrices, enhancing the design of interconnection networks with desirable topological properties.

Contribution

It defines the diagonal union operation for digraphs and demonstrates its use in constructing digraphs of unitary matrices for network design.

Findings

Diagonal union constructs digraphs of unitary matrices.

Digraphs from diagonal union are state split graphs.

Potential for improved interconnection network topologies.

Abstract

Many "good" topologies for interconnection networks are based on line digraphs of regular digraphs. These digraphs support unitary matrices. We propose the property "being the digraph of a unitary matrix" as additional criterion for the design of new interconnection networks. We define a composition of digraphs, which we call diagonal union. Diagonal union can be used to construct digraphs of unitary matrices. We remark that digraphs obtained via diagonal union are state split graphs, as defined in symbolic dynamics. Finally, we list some potential directions for future research.