Cospectral digraphs from locally line digraphs

A digraph Γ=(V,E)Γ=(V,E) is a line digraph when every pair of vertices u,v∈Vu,v∈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 Γ is a locally line digraph. In this paper we give a new method to obtain a digraph Γ′Γ′ cospectral with a given locally line digraph Γ with diameter D  , where the diameter D′D′ of Γ′Γ′ is in the interval [D−1,D+1][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.