Answered: 1.2.11. (−) Prove or disprove: If G is… | bartleby

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

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1.2.11. (−) Prove or disprove: If G is an Eulerian graph with edges e, f that share
vertex, then G has an Eulerian circuit in which e, f appear consecutively.
a

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31. In Example 2.35, describe all possible configurations of lights that can be obtained if we start with all the lights off.
3. [10 marks] Let Go = (V,E) and G₁ = (V,E₁) be two graphs on the same set of vertices. Let (V, EU E1), so that (u, v) is an edge of H if and only if (u, v) is an edge of Go or of G1 (or of both). H = (a) Show that if Go and G₁ are both Eulerian and En E₁ = Ø (i.e., Go and G₁ have no edges in common), then H is also Eulerian. (b) Give an example where Go and G₁ are both Eulerian, but H is not Eulerian.
7. [10 marks] Let G = (V,E) be a 3-connected graph with at least 6 vertices. Let C be a cycle in G of length 5. We show how to find a longer cycle in G. (a) Let x be a vertex of G that is not on C. Show that there are three C-paths Po, P1, P2 that are disjoint except at the shared initial vertex and only intersect C at their final vertices. (b) Show that at least two of P0, P1, P2 have final vertices that are adjacent along C. (c) Combine two of Po, P1, P2 with C to produce a cycle in G that is longer than C.
7. [10 marks] Let G = (V,E) be a 3-connected graph. We prove that for every x, y, z Є V, there is a cycle in G on which x, y, and z all lie. (a) First prove that there are two internally disjoint xy-paths Po and P₁. (b) If z is on either Po or P₁, then combining Po and P₁ produces a cycle on which x, y, and z all lie. So assume that z is not on Po and not on P₁. Now prove that there are three paths Qo, Q1, and Q2 such that: ⚫each Qi starts at z; • each Qi ends at a vertex w; that is on Po or on P₁, where wo, w₁, and w₂ are distinct; the paths Qo, Q1, Q2 are disjoint from each other (except at the start vertex 2) and are disjoint from the paths Po and P₁ (except at the end vertices wo, W1, and w₂). (c) Use paths Po, P₁, Qo, Q1, and Q2 to prove that there is a cycle on which x, y, and z all lie. (To do this, notice that two of the w; must be on the same Pj.)
Essentials of DISCRETE MATHEMATICS
1.2.17. (!) Let G,, be the graph whose vertices are the permutations of (1,..., n}, with two permutations a₁, ..., a,, and b₁, ..., b, adjacent if they differ by interchanging a pair of adjacent entries (G3 shown below). Prove that G,, is connected. 132 123 213 312 321 231
A wheel graph W₁, n ≥ 4 is the graph made from the cycle Cn-1 by adding an additional vertex vn (in the center of the cycle) and additional edges connecting Vn to each vertex of Cn-1. Describe the graph Wn and prove that Wn is Hamiltonian.
20. Determine whether the graphs G and H in Figure 12.76 are isomorphic. G: 21 U3 22 Us U5 27 U6 ug H: V4 V2 V6 V1 V8 Figure 12.76: The graphs in Exercise 20 V3 07 V5
ez V3 V2 V5 V1 V4 (a) Which edges are incident on v,? (b) Which vertices are adjacent to v? (c) Which edges are adjacent to e,?
(Discrete Math)
1.2.4. (-) Let G be a graph. For v € V(G) and e = E(G), describe the adjacency and incidence matrices of G-v and G-e in terms of the corresponding matrices for G.
9. AEFG has the vertices E(2, 6), F(8, 6), and g(6, 2). Find the orthocenter.

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