Cospectral digraphs from locally line digraphs

TL;DR

This paper introduces a new method to construct cospectral digraphs from locally line digraphs, preserving key properties and diameter within a small range, with applications to well-known digraph families.

Contribution

A novel technique for generating cospectral digraphs from locally line digraphs that maintains diameter and algebraic properties, extending the class of known cospectral constructions.

Findings

Constructed cospectral digraphs with diameter in [D-1,D+1]

Applied method to De Bruijn and Kautz digraphs

Preserved algebraic properties of original digraphs

Abstract

A digraph $\G=(V,E)$ is a line digraph when every pair of vertices $u,v\in V$ have either equal or disjoint in-neighborhoods. When this condition only applies for vertices in a given subset (with at least two elements), we say that $\G$ is a locally line digraph. In this paper we give a new method to obtain a digraph $\G'$ cospectral with a given locally line digraph $\G$ with diameter $D$, where the diameter $D'$ of $\G'$ is in the interval $[D-1,D+1]$. In particular, when the method is applied to De Bruijn or Kautz digraphs, we obtain cospectral digraphs with the same algebraic properties that characterize the formers.