The hamiltonian numbers in digraphs

In the paper, we study the hamiltonian numbers in digraphs. A hamiltonian walk of a digraph D is a closed spanning directed walk with minimum length in D. The length of a hamiltonian walk of a digraph D is called the hamiltonian number of D, denoted h(D). We prove that if a digraph D of order n is strongly connected, then $n\le h(D)\le \lfloor \frac{(n+1{)}^2}{4}\rfloor$ , and hence characterize the strongly connected digraphs of order n with hamiltonian number $\lfloor \frac{(n+1{)}^2}{4}\rfloor$ . In addition, we show that for each k with $4\le n\le k\le \lfloor \frac{(n+1{)}^2}{4}\rfloor$ , there exists a digraph with order n and hamiltonian number k. Furthermore, we also study the hamiltonian spectra of graphs.