Linear Algebra: A Modern Introduction

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.6: Applications And The Perron-frobenius Theorem

Problem 36EQ

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DISCRETE STRUCTURE

b) For the following pair of graphs GLUCONEOGENESIS and GLYCOLYSIS, show that
both graphs are isomorphic by
• Define the one-to-one and onto functions from the vertices of graph
GLUCONEOGENESIS and the vertices of graph GLYCOLYSIS.
• Construct the adjacency matrix.
GLYCOLYSIS
GLUCONEOGENESIS
ALDH2
ALDM
ACSSI
Acetate
Acetyl-
CoA
PBC
DLAT
Acetyl-
Acetate
CoA
Ругuvate
Pyruvate
ACSS2

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Draw a simple graph G with 7 vertices and δ(G) ≥ 3 that has no Euleriancycle, but has an Eulerian trail. Describe how you obtained this graph and whether the solution isunique (up to isomorphism). Justify your answer.
Q2 Let G be the following graph with 5 vertices: 3 (4. 1) Write the adjacency matrix of G 2) Enter your adjacency matrix on Pari (or other software) and use it to calculate the number of paths of length 10 starting at the vertex 1. How many of these paths end at vertex 1? Vertex 2, 3, 4, and 5?
Consider the graph below with vertex set {a,b,c,d,e,f,g}. a Give the adjacency matrix of the graph with lexiographically ordered rows and columns.
| Determine whether the two simple graphs with the following adjacency matrix are isomorphic or not? Check all criteria you know. (a) [0 0 1 0 0111 0], [0111001 00] (b) [0 110 0101 0011 0110 ], [0110 1101 1011 011 ]
Let A be the adjacency matrix of a complete graph K4.(a) Write down matrix A.(b) How many possible walks with length 2 are there from a (any) node to itself (e.g., from Node 2 to itself) (c) How many possible walks with length 3 are there from a (any) node to the other node (e.g., from Node 2 to Node 3 or Node 1 to Node 4)
For the directed graphs, draw the graph and show: A, A2 , A3, A4, A* , where A is the adjacency matrix for the edge set as a relation. (a) G1 = [{1, 2, 3, 4}]; (b) G2 = [{1, 2, 3, 4}]; Hint: Long matrix multiplications are not required, think about what for each of the matrices it means for a cell to contain a value of 1.
Determine if the graph shown below is Hamiltonian or Not * Hamiltonian Not Hamiltonian
Consider the graph G with • V(G) = {2,3,6} • e(G) = {a, b, c, d, e, f, g} •E(G) = {(a, [2,2]), (b, [3,3]), (c, [6,6]), (d, [2,6]), (e, [6,2]), (f, [3,6]), (g, [6,3])} Using edge connectivity, we can define the relation R = {(2, 2), (3, 3), (6, 6), (2, 6), (6, 2), (3, 6), (6,3)} Which of the following statements are true? [More than one statement may be true.] R is reflexive. R is symmetric. R is transitive. R is antisymmetric.
2. Let G be the following graph. V1 V2 V3 V5 VA (a) Let A be the adjacency matrix of G. Find A. (b) Using only A and its powers, determine how many walks of length EXACTLY 3 are there starting at 2 and ending at v4. Explain. (c) Using only A and its powers, determine how many walks of length AT MOST 3 are there starting at 2 and ending at v4. Explain.

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