Answered: 1.2.16. Let e be an edge appearing an… | 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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1.2.16. Let e be an edge appearing an odd number of times in a closed walk W. Prove
that W contains the edges of a cycle through c.

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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
1946 4. Give an example to show that if P is a (u, v)-path in a 2-connected graph G, then G does not necessarily contain a (u, v)-path Q internally-disjoint from P.
5. Let G = (V, E) be a finite, connected graph. Show that if no vertex of G has degree less than 2, then G contains a сycle.
Let G = (V, E) be a connected graph with a bridge e = uv. Prove that there exist two disjoint sets of vertices U,W whose union is V where any path of G from vertices of U to vertices of W contains e.
3 and 4
Prove that connecting two nodes u and v in a graph G by a new edge creates a new cycle if and only if u and v are in the same connected component of G.
Show that For n > 1 let Gn be the simple graph with vertex set V(Gn) = {1,2, ., n} in which two different vertices i and j are adjacent whenever j is a multiple of i or i is a multiple of j. For what n is Gn planar? ...1
5.1.21. Suppose that every edge of a graph G appears in at most one cycle. Prove that every block of G is an edge, a cycle, or an isolated vertex. Use this to prove that x (G) < 3.
a. Let V = {x, y, z, u, v}. How many distinct graphs are there on the vertex set V with exactly 6 edges? (Note that for graphs G₁ = (V, E₁) and G₂ = (V, E₂) to be distinct, they need only have E₁ # E2, but they may be isomorphic.) b. In the graph below, give an example of a trail that is not a path. I Y U Z
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.
3. Suppose that G = (V, E) is a directed graph. A vertex w € V is called reachable from a vertex v E V if there is a directed path from v to w. The vertices v and w are mutually reachable if there is both a directed path from v to w and a directed path from w to v in G. Show that if G = (V, E) is a directed graph and that u, v and w are vertices in V for which u and v are mutually reachable and v and w are mutually reachable, then u and w are mutually reachable. 4. Find the number of paths of length n between two different vertices in K4 if n is • 2 ● 3

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