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.
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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.
1.2.12. (-) Convert the proof at 1.2.32 to an procedure for finding an Eulerian circuit in a connected even graph.
1.2.7. (-) Prove that a bipartite graph has a unique bipartition (except for interchang- ing the two partite sets) if and only if it is connected.
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 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. Ꮖ (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 x and only intersect C at their final vertices. (b) Show that at least two of Po, P1, P2 have final vertices that are adjacent along C.
3.1.11. Let C and C' be cycles in a graph G. Prove that CaC' decomposes into cycles.
1.2.18. (!) Let G be the graph whose vertex set is the set of k-tuples with elements in (0, 1), with x adjacent to y if x and y differ in exactly two positions. Determine the number of components of G.
1.1.30. Let G be a simple graph with adjacency matrix A and incidence matrix M. Prove that the degree of vi is the ith diagonal entry in A2 and in MMT. What do the entries in position (i, j) of A2 and MMT say about G?
8. Answer these two questions:
4.1.12. Let n, k be positive integers with n even, k odd, and n > k > 1. Let G be the k- regular simple graph formed by placing n vertices on a circle and making each vertex adjacent to the opposite vertex and to the (k – 1)/2 nearest vertices in each direction. Prove that < (G) = k. (Harary (1962a])
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