The classical game of Peg Solitaire has uncertain origins, but was certainly popular by the time of LOUIS XIV, and was described by LEIBNIZ in 1710. The modern mathematical study of the game dates to the 1960s, when the solitaire cone was first described by BOARDMAN and CONWAY. Valid inequalities over this cone, known as pagoda functions, were used to show the infeasibility of various peg games...

We show that the adjacency matrix M of the line digraph of a d-regular digraph D on n vertices can be written as M = AB, where the matrix A is the Kronecker product of the all-ones matrix of dimension d with the identity matrix of dimension n and the matrix B is the direct sum of the adjacency matrices of the factors in a dicycle factorization of D.