Answered: 5. Let k be a fixed positive integer… | 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

Problem 74EQ

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5. Let k be a fixed positive integer and let G = (V,E) be a loop-free undirected graph, where
deg(v) ≥ k for all vЄ V. Prove that G contains a path of length k.

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Let P₁ and P₂ be two paths of maximum length in a connected graph G. Prove that P₁ and P2 have a common vertex.
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)}.
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7. a) How many different paths of length 2 are there in the undirected graph G in Fig. 11.43? b) Let G = (V, E) be a loop-free undirected graph, where V= {U, v2. ..., v) and deg(v,) = d,, for all I sisn. How many different paths of length 2 are there in G? Figure 11.43
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5. If G is a simple graph with d(v) 2 k, Vv € V (G), then G contains a path of length at least k. If k2 2, then G contains a cycle of length k+1.
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14. Let G be the graph with vertices v1, v2, and v3 as shown below. What is the number of walks of length 3 from v2 to v3? v2 V3
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