On the Cayley Digraphs That Are Patterns of Unitary Matrices

Study the relationship between unitary matrices and their patterns is motivated by works in quantum chaology (see, e.g., [KS99]) and quantum computation (see, e.g., [M96] and [AKV01]). We prove that if a Cayley digraph is a line digraph then it is the pattern of a unitary matrix. We prove that for any finite group with two generators there exists a set of generators such that the Cayley digraph with respect to such a set is a line digraph and hence the pattern of a unitary matrix. 1. Definitions The following standard definitions will be central: Definition 1. Let G = 〈S〉 be a finite group. The ( right) Cayley digraph Cay (G, S) of G with respect to S is the digraph whose vertex set is G, and whose arc set is the set of all ordered pairs {(g, gs) : g ∈ G and s ∈ S}. Definition 2. Let D = (V, A) be a digraph. The line digraph L (D) of D is the digraph whose vertex set is A (D), and ((u, v) , (w, z)) ∈ A (L (D)) if and only if v = w, where u, v, w, z ∈ V (D) and (u, v) , (w, z) ∈ A (D). Definition 3. Let M be a square matrix M of size n. A digraph D is said to be the pattern of M , if D is on n vertices and, for every u, v ∈ V (D), (u, v) ∈ A (D) if and only if the entry Muv is nonzero. Definition 4. A square matrix U with complex entries is said to be unitary if it is nonsingular and U U = I, where U † and I denote the adjoint of U and the identity matrix, respectively. 2. On the Cayley digraphs that are patterns of unitary matrices Denote by M (D) the adjacency matrix of a digraph D. Lemma 1. If a regular digraph is a line digraph then it is the pattern of a unitary matrix. Date: June, 2002. 1991 Mathematics Subject Classification. Primary 05C10.