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

For a digraph D  , let L(D)L(D) and S(D)S(D) denote its line digraph and subdivision digraph, respectively. The motivation of this paper is to solve the digraph equation L(S(D))=S(L(D))L(S(D))=S(L(D)). We show that L(S(D))L(S(D)) and S(L(D))S(L(D)) are cospectral if and only if D   and L(D)L(D) have the same number of arcs. Further, we characterize the situation that L(S(D))L(S(D)) and S(L(D))S(L(D)) are isomorphic. Our approach introduces the new notion, the proper image D*D* of a digraph DD, and a new type of connectedness for digraphs. The concept D*D* 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, L(D)L(D) is connected if and only if D*D* is connected; L(D)L(D) is functional (contrafunctional) if and only if D*D* is functional (contrafunctional). Some related results are also presented.