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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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3. Let G = (V, E) be a loop-free undirected n-regular graph with |V | ≥ 2n + 2. Prove that G
(the complement of G) has a Hamilton cycle.

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Let G be a simple connected graph with n vertices and 1/2(n-1)(n-2)+2 edges. Use Ore's theorem to prove that G is Hamiltonian.
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.
10. Let G = (V, E) be a loop-free connected planar graph. If G is isomorphic to its dual and|V | = n, what is |E|?
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.
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.
Let Vn be the set of connected graphs having n edges, vertex set [n], and exactly one cycle. Form a graph Gn whose vertex set is Vn. Include {gn, hn} as an edge of Gn if and only if gn and hn differ by two edges, i.e. you can obtain one from the other by moving a single edge. Tell us anything you can about the graph Gn. For example, (a) How many vertices does it have? (b) Is it regular (i.e. all vertices the same degree)? (c) Is it connected? (d) What is its diameter?
Let G be a connected graph of order n and size n. Prove that G contains a single cycle.
Let G (V, E) be a graph such that = • V ≥ 3, G has exactly 2 leaves, and in G all the non-leaf vertices have degree 3 or more. Prove that G has at least one cycle. You are not required to draw anything in your proof.
I have to prove the following lemma: “Let G be a 2-connected graph. If e and f are parallel edges in G, then G\e is 2-connected.” I was thinking about proving it by showing that there exists a cycle betweent two adjacent vertices u and v such that if we were to delet one of the edges, either e or f, we would remain in a 2-connected graph. I'm not sure if this is correct and even how to word it properly. Please help me with this prove, thank you :)
If G = (V, E) has n > 2 vertices and no self-loops, show that there exist two vertices v # w such that deg(v) = deg(w). Present a counterexample, if G is allowed to have self-loops.
The following question are mostly on the topic of paths, cycles and connectedness - (a) Let G be a graph of order n ≥ 2 such that (G) ≥ 1/1/2(n − 1). Show that any two non-adjacent vertices in G have a common neighbor. (b) Let G be an (n,m) graph such that m> (21). Show that G is connected. (c) Let G be a connected graph that is not complete. Show that there exists three vertices u, v, w in G such that uv = E(G), vw = E(G), but uw & E(G).
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