The concept of diameter in exponents of symmetric primitive graphs (Q2761051)

A directed graph \(G\) is called primitive, if there exists an integer \(k\) such that for any two vertices \(u,v\) of \(G\) there exists a walk from \(u\) to \(v\) of length \(k\). The least number \(k\) with this property is the exponent \(\exp (G)\) of \(G\). It is proved that if a primitive digraph \(G\) is symmetric, then \(\exp (G)\leq 2d\), where \(d\) is the diameter of \(G\). Symmetric primitive digraphs \(G\) with \(\exp (G)=2d\) and ones with \(\exp (G) = n\), where \(n\) is the number of vertices, are characterized.