Consider the following directed graph G(V,E): e1 V2 e4 V3 V1 e5 e3 e6 e2 e7...

  

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Consider the following directed graph G(V,E): e1 V2 e4 V3 V1 e5 e3 e6 e2 e7 V4 V5 (i) Find the adjacency matrix A and incidence matrix B. (ii) Consider the undirected version of the graph and find its adjacency matrix C. (iii) Show that BT1 = 0, where 1 and 0 represent the all-ones and zero vectors, respectively. (iv) Let L = BBT. Show that L = D-C, where D = diag(deg(v1), ..., deg(vs)). (v) Find det(L). (vi) Show that all eigenvalues of L are nonnegative. Consider the following directed graph G(V,E): e1 V2 e4 V3 V1 e5 e3 e6 e2 e7 V4 V5 (i) Find the adjacency matrix A and incidence matrix B. (ii) Consider the undirected version of the graph and find its adjacency matrix C. (iii) Show that BT1 = 0, where 1 and 0 represent the all-ones and zero vectors, respectively. (iv) Let L = BBT. Show that L = D-C, where D = diag(deg(v1), ..., deg(vs)). (v) Find det(L). (vi) Show that all eigenvalues of L are nonnegative.