4. Given any directed graph D, we can transform it in several ways to obtain: • an undirected graph un(D) by "forgetting orientations of the edges"; • an opposite directed graph op(D) by "reversing...

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Extracted text: 4. Given any directed graph D, we can transform it in several ways to obtain: • an undirected graph un(D) by "forgetting orientations of the edges"; • an opposite directed graph op(D) by "reversing the orientations of the edges"; • a loopy (di)graph l(D) by deleting every edge that is not a loop. Recall that for a graph or digraph X we denote by Ax its adjacency matrix. Given an arbitrary digraph D, prove the following statements. (a) Ag(D) = A, where A° denotes the main diagonal submatrix: A° (i, i) = A(i, i) and A° (i, j) = 0 whenever i + j. (b) Aop(D) = A5, where AT is the transpose matri defined by A" (i, j) = A(j, i). (c) Aun(D) = Ap + Aop(D) – A«D).