The exponent and circumdiameter of primitive digraphs (Q1763834)

The exponent of a primitive digraph \(D\) is the smallest \(m\) such that for each ordered pair \((u,v)\) of not necessarily distinct vertices, there exists a \(u\)-\(v\) walk of length \(m\). The circumdiameter of \(D\) is the maximum over all ordered pairs \((u,v)\) of not necessarily distinct vertices of the length of a shortest \(u\)-\(v\) walk that intersects cycles of all lengths. It is known that the exponent of \(D\) is less than or equal to the sum of its Frobenius-Schur index and the circumdiameter. The paper gives several new sufficient conditions and families of digraphs for which equality holds in the above upper bound. Additional sufficient conditions for equality in the above upper bound for the exponent of \(D\) and a new upper bound for the circumdiameter are given for digraphs with large exponent. The paper also presents circumdiameter and bounds on the exponent for the digraph of a Leslie matrix.