Local exponents of primitive digraphs

A digraph G = (V, E) is primitive if, for some positive integer k, there is a u → v walk of length k for every pair u, v of vertices of V. The minimum such k is called the exponent of G, denoted exp(G). The local exponent of G at a vertex u V, denoted expG(u), is the least integer k such that there is a u → v walk of length k for each v V. Let V = {{1, 2, …, n}}. Following Brualdi and Liu, we order the vertices of V so that expG(1) expG(2) … expG(n) = exp(G). It is known that expG(k) s(n − 2) + k for all k, 1 k n. The problem of characterizing the exponent set ESn = {{exp(G) : G Pn}}, where Pn is the set of all primitive digraphs of order n, has been completely settled. We define the ith local exponent set ESn(i) := {{expG(i) : G Pn}} for each i, 1 i n, and show that ESn(1) has a characterization which closely parallels that of ESn(n) = ESn. We conjecture that, except for minor changes, there are similar formulae for the ESn(i) for all 1 i n.