Line Digraphs and Coined Quantum Random Walks

In this note, we give a characterization of the adjacency matrix of the line digraph of a regular digraph and we discuss a generalization.On the light of the characterization given, we remark that the underlying digraph of a coined quantum random walk is the line digraph of a regular digraph. 1. A characterization of the adjacency matrix of the line digraph of a regular digraph 1.1. Set-up. The notion of line digraph has been introduced by Harary and Norman [HN60] in 1960. A classic survey on line graphs and digraphs is [HB78]; a recent one is [P95]. Line digraphs are used in the design and analysis of interconnection networks (see e.g. [FYA84]). Line digraphs are also used in algorithms for DIRECTED MAX-CUT [CE90] and TRAVELLING SALESMAN [GKWS98]. Definition 1 (Line digraph). The line digraph −→ LD of a digraph D = (V,A) is the digraph whose vertex set is A (D), and ((vi, vj) , (vk, vl)) ∈ A (−→ LD ) if and only if vj = vk, where vi, vj , vk, vl ∈ V (D) and (vi, vj) , (vk, vl) ∈ A (D). Iteratively, −→ L D = −→ L k−1 −→ LD, and −→ L D is called k-line digraph of D. Example 1. Let denote byKd the complete symmetric digraph. Let denote byK + d the complete symmetric digraph with a loop at each vertex. The de Bruijn digraph B (d, k) and the Kautz digraph K (d, k) can be defined as follows (see [FYA84]): B (d, k) = −→ L Kd and K (d, k) = −→ L K d . We remark that the line digraph of an Eulerian digraph is: • Hamiltonian (see e.g. [HB78]); • The pattern of a unitary matrix (see [S02]), that is, there are unitary matrices such that their ij-th entry is nonzero, if and only if (vi, vj) is an arc. It might be interesting to remark that: • If D is strong, k-regular and on n-vertices, then −→ LD is strong, k-regular, and on n · k vertices (see e.g. [HB78], Theorem 7.4). • The eigenvalues of −→ LD are the eigenvalues of D plus a zero eigenvalue with algebraic multiplicity n (k − 1) (see e.g. [R01], Theorem 3); Date: September 2002. 1991 Mathematics Subject Classification. Primary 05C50.