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- 3. Consider a network (G, w). Let c ER, and let m: E(G) → R such that m(e) = w(e) + c for all e Є E(G). (a) Show that T is a minimum spanning tree of (G, w) if and only if it is a minimum spanning tree of (G,m).3. Consider a network (G, w). Let c € R, and let m : E(G) → R such that m(e) w(e) + c for all e = E(G). = (a) Show that T is a minimum spanning tree of (G, w) if and only if it is a minimum spanning tree of (G,m). (b) What is the analogous claim for shortest paths in (G, w) and (G, m)? Prove this claim or provide a counterexample showing that it is not true.Give a simple example of a connected graph such that an edge (u, v) is a light edge for some cut, but there exists a minimum spanning tree that does not contain (u, v).
- Let V = Mnxn(F). (a) Suppose that {M1,..., Mt} C Mnxn(F) is a spanning set. Prove that {M{,.…, M£} is also a spanning set. (Hint: Express At as a linear combination of the M;'s and use transpose carefully.) (b) Prove that if B = {M11, M12, -.. , Mnn} is any basis of V, then B' = {M, M, ..., Min} is also a basis of V.Let G₁ and G₂ be two distinct trees with the same vertex-set V. Prove that d(G₁)=d(G₂) if (G₁-x) = (G₂-x) for any x EV.If G = (V, E) has n > 2 vertices and no self-loops, show that there exist two vertices v # w such that deg(v) = deg(w). Present a counterexample, if G is allowed to have self-loops.
- 2. Let G be a graph and e € E(G). Let H be the graph with V(H) = V(G) and E(H) = E(G)\{e}. Then e is a bridge of G if H has a greater number of connected components than G. (b) Assume that G is connected and that e is a bridge of G with endpoints u and v. Show that H has exactly two connected components H₁ and H₂ with u € V(H₁) and v € V (H₂). To this end, you may want to consider an arbitrary vertex w € V (G) and use a u-w-path in G to construct a u-w-path or a v-w-path in H.Let G be a connected graph with at least one edge and F ⊆ E(G) be an edge cut. Prove that F is a minimal edge cut if and only if G − F contains exactly two connected components.Let G = (V, E) be a finite bipartite graph with bipartition (A, B) where |A| = |B|. We say that G satisfies the marriage condition iff for every SCA, ING(S) ≥ |S| where NG(S) = {be B: (3a € S)({a, b} = E)} It should be clear that if G has a perfect matching, then it satisfies the marriage condition. Hall's theorem says that the converse is also true.
- Draw the directed graphs of the relation.Let G = (V, E) be a graph with vertex-set V = {1, 2, 3, 4, 5} and edge-setE = {(1, 2),(3, 2),(4, 3),(1, 4),(2, 4),(1, 3)}.(a) Draw the graph.Find (b) maximal degree, i.e. ∆(G),(c) minimal degree, i.e. δ(G),(d) the size of biggest clique, i.e. ω(G),(e) the size of biggest independent set, i.e. α(G), ter(f) the minimal number of colours needed to color the graph, i.e. χ(G).In chess. A “knight’s move” consists of two squares either vertically or horizontally and then one square is a perpendicular direction. Depending on where the knight is situated, he has a minimum mobility of two moves—when in a corner—and a maximum mobility of eight moves. Let C be a graph with v=64, its vertices corresponding to the squares of a chessboard. Let two vertices of C be joined by an edge whenever a knight can go from one of the corresponding squares to the other in one move. Does C have an Euler Walk? Explain, but you do not have to draw C to answer.