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 ((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) A(D) = A, where A° denotes the main diagonal submatrir: A°(i, i) = A(i, i) and A° (i, j) = 0 whenever i + j. (b) Aop(D) = A5, where A" is the transpose matri defined by A" (i, j) = A(j, i). (c) Aun(D) = Ap + Aop(D) – A(D). %3D

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Ornithologists have determined that some species of birds tend to avoid flights over large bodies of water during daylight hours. It is believed that more energy is required to fly over water than land because air generally rises over land and falls over water during the day. island 4 km 16 km W L D nest @ A bird with these tendencies is released from an island that is 4 km from the nearest point B on a straight shoreline, flies to a point C on the shoreline, and then flies along the shoreline to its nesting area D. Points B and D are 16 km apart. Assume that the bird instinctively chooses a path that will minimize its energy expenditure. (Round your answers to two decimal places.) (a) In general, if it takes 1.2 times as much energy to fly over water as land, to what point C should the bird fly in order to minimize the total energy expended in returning to its nesting area? km from B (b) Let W and L denote the energy (in joules) per kilometer flown over water and land, respectively.…

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