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 minimumspanning tree of (G,m).

Ans-3)(a) To show that T is a minimum spanning tree of (G, w) if and only if it is a minimum…

Answered: 3. Consider a network (G, w). Let c €…

Linear Algebra: A Modern Introduction

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ISBN:9781285463247

Author:David Poole

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Chapter4: Eigenvalues And Eigenvectors

Section4.6: Applications And The Perron-frobenius Theorem

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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.
Let (u, v) be a minimum-weight edge in a connected graph G. Show that (u, v) belongs to some minimum spanning tree of G.
= 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.
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).
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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).
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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).