On commutativity of two unary digraph operations: Subdividing and line-digraphing Original Research Article

For a digraph D, let image and image denote its line digraph and subdivision digraph, respectively. The motivation of this paper is to solve the digraph equation image. We show that image and image are cospectral if and only if D and image have the same number of arcs. Further, we characterize the situation that image and image are isomorphic. Our approach introduces the new notion, the proper image image of a digraph image, and a new type of connectedness for digraphs. The concept image plays an important role in the main result of this paper. It is also useful in other aspects of the study of line digraphs. For example, image is connected if and only if image is connected; image is functional (contrafunctional) if and only if image is functional (contrafunctional). Some related results are also presented.