Abstract

Let G be a graph. An edge orientation of G is called smooth
if the indegree and the outdegree of every vertex differ by
at most one. In this paper, we show that if G is a 2edgeconnected nonbipartite graph with δ(G) ≥ 3, then G has a
smooth primitive orientation. Among other results, using the
spectral radius of digraphs, we show that if D1 is a primitive
regular orientation and D2 is a nonregular orientation of a
given graph, then for sufficiently large t, the number of closed
walks of length t in D1 is more than the number of closed
walks of length t in D2.
