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Linear Algebra: A Modern Introduction

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter3: Matrices

Section3.7: Applications

Problem 80EQ

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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.

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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 € 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.
Prove (Menger) if x, y are vertices of a graph G and xy e E(G), then the minimum size of an x,y-cut equals the maximum number of pairwise internally disjoint x,y-paths
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).
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.
5. Let G = (V, E) be a graph with vertex-set V = {1,2,3,4} and edge-set E = {(1, 2), (3, 2), (4, 3), (1, 4), (2,4)}. (a) Draw the graph. Find (b) maximal degree, i.e. A(G), (c) minimal degree, i.e. 8(G), (d) the size of biggest clique, i.e. w(G), (e) the size of biggest independent set, i.e. a(G), ter (f) the minimal number of colours needed to color the graph, i.e. x(G).
Let G be a simple graph with nonadjacent vertices v and w, and let G+e denote the simple graph obtained from G by creating a new edge, e, joining v and w. Prove that x(G) = min{x(G+e), x((G+e) 4e)}.
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.
Give a path of maximum length in G.
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).
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.
Suppose a graph G has 22 vertices. What is the minimum number of EDGES that G must contain if Dirac's Theorem can be used to guaranteea Hamiltonian Grat in G?

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