Two results concerning distance-regular directed graphs (Q2712507)

The paper deals with distance-regular oriented graphs. Let \(\Gamma\) be a distance-regular oriented graph with diameter \(d\) and girth \(g\). Then either \(g=d+1\) (and the digraph is said to be short) or \(g=d\) (and the digraph is said to be long). Every long graph can be constructed from an associated short graph (Damerell). A short distance-regular digraph which is not a directed circuit, has girth at most 8 and examples with girth 3 or 4 are known. Theorem A. Let \(C\) be the intersection matrix of a short distance-regular digraph with girth \(g\) and valency \(k\). Then \(C\) has \(g\) distint eigenvalues, and (i) if \(g\) is odd, then \(C\) has exactly one real eigenvalue \(k\) and (ii) if \(g\) is even, then \(C\) has exactly two real eigenvalues, one of which is \(k\) and the other is a negative number. Theorem B. Let \(\Gamma\) be a short distance-regular digraph with valency \(k\) and \(n\) vertices. (i) If \(k=1\), then \(\Gamma\) is a directed \(n\)-circuit and is primitive if and only if \(n\) is a prime. (ii) If \(k>1\), then \(\Gamma\) is primitive.