 5.1: For the graph shown in Fig. 29, (a) give the vertex set. (b) give t...
 5.2: For the graph shown in Fig. 30, (a) give the vertex set. (b) give t...
 5.3: For the graph shown in Fig. 31, (a) give the vertex set. (b) give t...
 5.4: For the graph shown in Fig. 32, (a) give the vertex set. (b) give t...
 5.5: Consider the graph with vertex set 5K, R, S, T, W6 and edge list RS...
 5.6: Consider the graph with vertex set 5A, B, C, D, E6 and edge list AC...
 5.7: Consider the graph with vertex set 5A, B, C, D, E6 and edge list AD...
 5.8: Consider the graph with vertex set 5A, B, C, X, Y, Z6 and edge list...
 5.9: (a) Give an example of a connected graph with six vertices such tha...
 5.10: a) Give an example of a connected graph with eight vertices such th...
 5.11: Consider the graph in Fig. 33. (a) Find a path from C to F passing ...
 5.12: Consider the graph in Fig. 33. (a) Find a path from D to E passing ...
 5.13: Consider the graph in Fig. 33. (a) Find all circuits of length 1. (...
 5.14: Consider the graph in Fig. 34. (a) Find all circuits of length 1. (...
 5.15: List all the bridges in each of the following graphs: (a) the graph...
 5.16: List all the bridges in each of the following graphs: (a) the graph...
 5.17: Consider the graph in Fig. 35. (a) Find the largest clique in this ...
 5.18: Consider the graph in Fig. 36. (a) Find the largest clique in this ...
 5.19: Figure 37 shows a map of the downtown area of the picturesque hamle...
 5.20: Figure 38 is a map of downtown Royalton, showing the Royalton River...
 5.21: A night watchman must walk the streets of the Green Hills subdivisi...
 5.22: A mail carrier must deliver mail on foot along the streets of the G...
 5.23: Six teams 1A, B, C, D, E, and F2 are entered in a softball tourname...
 5.24: The Kangaroo Lodge of Madison County has 10 members 1A, B, C, D, E,...
 5.25: Table 3 summarizes the Facebook friendships between a group of eigh...
 5.26: The Dean of Students office wants to know how the seven general edu...
 5.27: Figure 40 shows the downtown area of the small village of Kenton. T...
 5.28: Figure 40 shows the downtown area of the small village of Kenton. A...
 5.29: In Exercises 29 through 34 choose from one of the following answers...
 5.30: In Exercises 29 through 34 choose from one of the following answers...
 5.31: In Exercises 29 through 34 choose from one of the following answers...
 5.32: In Exercises 29 through 34 choose from one of the following answers...
 5.33: In Exercises 29 through 34 choose from one of the following answers...
 5.34: In Exercises 29 through 34 choose from one of the following answers...
 5.35: Find an Euler circuit for the graph in Fig. 47. Show your answer by...
 5.36: Find an Euler circuit for the graph in Fig. 48. Show your answer by...
 5.37: Find an Euler path for the graph in Fig. 49. Show your answer by la...
 5.38: Find an Euler path for the graph in Fig. 50. Show your answer by la...
 5.39: Find an Euler circuit for the graph in Fig. 51. Use B as the starti...
 5.40: Find an Euler circuit for the graph in Fig. 52. Use S as the starti...
 5.41: Suppose you are using Fleurys algorithm to find an Euler circuit fo...
 5.42: Suppose you are using Fleurys algorithm to find an Euler circuit fo...
 5.43: Find an optimal eulerization for the graph in Fig. 54.
 5.44: Find an optimal eulerization for the graph in Fig. 55.
 5.45: Find an optimal eulerization for the graph in Fig. 56. FiGurE 56
 5.46: Find an optimal eulerization for the graph in Fig. 57. FiGurE 57
 5.47: Find an optimal semieulerization for the graph in Fig. 56. You are...
 5.48: Find an optimal semieulerization for the graph in Fig. 57. You are...
 5.49: Find an optimal semieulerization of the graph in Figure 56 when A ...
 5.50: Find an optimal semieulerization of the graph in Figure 57 when A ...
 5.51: Find an optimal semieulerization of the graph in Figure 56 when B ...
 5.52: Find an optimal semieulerization of the graph in Fig. 57 when A an...
 5.53: A security guard must patrol on foot the streets of the Green Hills...
 5.54: A mail carrier must deliver mail on foot along the streets of the G...
 5.55: This exercise refers to the Fourth of July parade problem introduce...
 5.56: This exercise refers to the Fourth of July parade problem introduce...
 5.57: Assume you want to trace the diagram of a basketball court shown in...
 5.58: (a) Explain why in every graph the sum of the degrees of all the ve...
 5.59: If G is a connected graph with no bridges, how many vertices of deg...
 5.60: Regular graphs. A graph is called regular if every vertex has the s...
 5.61: Complete bipartite graphs. A complete bipartite graph is a graph ha...
 5.62: Consider the following game. You are given N vertices and are requi...
 5.63: Consider the following game. You are given N vertices and allowed t...
 5.64: Figure 60 shows a map of the downtown area of the picturesque hamle...
 5.65: A policeman has to patrol on foot the streets of the subdivision sh...
 5.66: Describe an optimal route for the photographer if the route must st...
 5.67: Describe an optimal route for the photographer if the route must st...
 5.68: Describe an optimal route for the photographer if the route must st...
 5.69: This exercise comes to you courtesy of Euler himself. Here is the q...
 5.70: Suppose G is a connected graph with N vertices, all of even degree....
 5.71: Suppose G is a connected graph with N  2 even vertices and two odd...
 5.72: Suppose G is a disconnected graph with exactly two odd vertices. Ex...
 5.73: Suppose G is a simple graph with N vertices 1N 22. Explain why G mu...
 5.74: Kissing circuits. When two circuits in a graph have no edges in com...
 5.75: Hierholzers algorithm. Hierholzers algorithm is another algorithm f...
Solutions for Chapter 5: The Mathematics of Getting Around
Full solutions for Excursions in Modern Mathematics  8th Edition
ISBN: 9781292022048
Solutions for Chapter 5: The Mathematics of Getting Around
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 5: The Mathematics of Getting Around includes 75 full stepbystep solutions. This textbook survival guide was created for the textbook: Excursions in Modern Mathematics, edition: 8. Since 75 problems in chapter 5: The Mathematics of Getting Around have been answered, more than 13340 students have viewed full stepbystep solutions from this chapter. Excursions in Modern Mathematics was written by and is associated to the ISBN: 9781292022048.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)ยท(b  Ax) = o.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Solvable system Ax = b.
The right side b is in the column space of A.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.