Answered: 1.2.10. (-) Prove or disprove: a) Every… | 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 80EQ

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1.2.10. (-) Prove or disprove:
a) Every Eulerian bipartite graph has an even number of edges.
b) Every Eulerian simple graph with an even number of vertices has an even num-
ber of edges.

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I want this to be considered as a Advanced Math question pls. . Consider a graph G which is a complete bipartite graph. The graph G is defined as K(3,4), meaning it has two sets of vertices, with 3 vertices in one set and 4 in the other. Every vertex in one set is connected to every vertex in the other set, but there are no connections within a set. Calculate the number of edges in graph G. Also, determine if the graph G contains an Euler path or circuit, and justify your answer.
(Note: Please provide a precise answer and explain briefly which is not provided on Chegg or Bartleby.)
2. Prove that any graph has at least two vertices with the same degree. A complete bipartite graph on ( m, n) vertices, is a simple graph whose vertices can be divided into two distinct, non-overlapping sets (that is, suppose V has m vertices and W has n vertices) in such a way that there is exactly one edge from each vertex of V to each vertex of W , there is no edge from any one vertex of V to any other vertex of V , and there is no edge from any one vertex of W to any other vertex of W. Use ways to select the edges to show that this graph has m.n edges Use combinations to show that the number of edges on a complete graph is n(n-1)/2 (NOTE: Please elaborate on the answver and explain. Please do not copy-paste the answer from the internet or from Chegg.)
3. (a) Is it possible to have a 4-regular graph with 15 vertices? If no, explain why. If yes, construct such a graph. (b) The degree of every vertex of a graph G is one of three consecutive integers. If, for each of the three consecutive integers r, the graph G contains exactly r vertices of degree r, prove that two-thirds of the vertices of G have odd degree.
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
2.12 Prove that a 3-regular graph has a cut vertex if, and only if, it has some bridge.
(b) Suppose G is a simple connected graph with 12 vertices and 16 edges. Suppose 4 of its vertices are degree 1, and 3 of its vertices are degree 2. Prove that G is planar. (Hint: Kuratowski) (c) Let G be any simple connected planar graph with n vertices and e edges. Suppose there are exactly y vertices of degree 2. Assume that n - y > 3. Prove that e < 3n - y - 6. (Hint: Explain why the degree-2 vertices can be erased, and how to take care of any resulting loops or multiple edges.) (d) Suppose that a connected simple graph G' has exactly 10 vertices of degree 4, 8 vertices of degree 5, and all other vertices have degree 7. Find the maximum possible number of degree-7 vertices G could have, so that G would still be planar.
7. Answer these two questions:(a) Find all the nonisomorphic complete bipartite graphs G = (V, E), where |V | = 6.(b) How many nonisomorphic complete bipartite graphs G = (V, E), satisfy |V | = n ≥ 2?
Let G be a simple graph with exactly 11 vertices. Prove that G or its complement G must benon-planar. Hint: The maximum number of edges in a planar graph with n vertices is 3n − 6.Please write in complete sentences, include all details, showall of your work, and clarify all of your reasoning.
A graph is bipartite if its vertex set can be partitioned into two sets V₁ and V2 such all edges are between V₁ and V2 (i.e. there are no edges joining vertices inside V₁, and the same for V2). (a) Draw a bipartite graph with 5 vertices and 5 edges. (b) What is the maximum number of edges for a bipartite graph with 2n vertices (suppose n > 1)?
Which one is correct?
4. [10 marks] Find both a matching of maximum size and a vertex cover of minimum size in the following bipartite graph. Prove that your answer is correct. ย ພ

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