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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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6. Let G be an m-ary tree of height h. Prove that G has
(a) at most m leaves, and
(b) at most
m" vertices.
i=0

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1a. Prove: every tree with n ≥ 2 vertices has at least 2 leaves. (3 pt) 1b. Let T be a tree. Prove: if all vertices have degree either 1 or at least 4, then T has at least 2(n + 1)/3 leaves. (4 pt)
7. Prove: "If T is a tree of order n and size m, then m = n – 1." Give a concrete example to elaborate your proof. %3D End -
1. (Prunes) Prove that if G is a tree with a vertex of degree d, then it has at least d leaves. 2. (Two paths) Prove that in every tree T, any two paths of marimum length have a node in common. (I.e. if vo, v1, . .., Vk and wo, w1, ..., Wk are paths of maximum length in a tree T, then there are integers i, je {0, 1, ..., k} such that v; = wj.)
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3. Let T be a tree all of whose vertices have degree 1 or 3. Suppose T has n leaves (a) Show that T has m = n – 2 vertices of degree 3. (b) Show that if n > 4, there is some internal vertex which is adjacent to two leaves. (Hint: Consider the subgraph of T on the internal vertices)
6. [10 marks] Let T be a tree with n ≥ 2 vertices and leaves. Let BL(T) denote the block graph of T. (a) How many vertices does BL(T) have? (b) How many edges does BL(T) have? Prove that your answers are correct.
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c) A 3-ary tree of height 3 with exactly 19 leaf vertices.
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.
4a Let n 2 4. WVhat is the maximum possible number of edges in a graph with n vertices and n - 2 connected components? Prove your answer. 4b How many different undirected graphs can be formed with vertex set V = {1,2,3, 4}? 2}) and (V, {2 – 3}) as two different (The vertices are distinguishable, so we count (V,{1 graphs, for example.)
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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.)

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