 7.1: A computer lab has seven computers labeled A through G. The connect...
 7.2: The following is a list of the electrical power lines connecting ei...
 7.3: Consider the network shown in Fig. 20. (a) How many degrees of sepa...
 7.4: Consider the network shown in Fig. 21. (a) How many degrees of sepa...
 7.5: Consider the tree shown in Fig. 22 on the next page. (a) How many d...
 7.6: Consider the tree shown in Fig. 23. (a) How many degrees of separat...
 7.7: In Exercises 7 through 20 you are given information about a network...
 7.8: In Exercises 7 through 20 you are given information about a network...
 7.9: In Exercises 7 through 20 you are given information about a network...
 7.10: In Exercises 7 through 20 you are given information about a network...
 7.11: In Exercises 7 through 20 you are given information about a network...
 7.12: In Exercises 7 through 20 you are given information about a network...
 7.13: In Exercises 7 through 20 you are given information about a network...
 7.14: In Exercises 7 through 20 you are given information about a network...
 7.15: In Exercises 7 through 20 you are given information about a network...
 7.16: In Exercises 7 through 20 you are given information about a network...
 7.17: In Exercises 7 through 20 you are given information about a network...
 7.18: In Exercises 7 through 20 you are given information about a network...
 7.19: In Exercises 7 through 20 you are given information about a network...
 7.20: In Exercises 7 through 20 you are given information about a network...
 7.21: Consider the network shown in Fig. 24. (a) Find a spanning tree of ...
 7.22: Consider the network shown in Fig. 25. (a) Find a spanning tree of ...
 7.23: Consider the network shown in Fig. 26. (a) Find a spanning tree of ...
 7.24: Consider the network shown in Fig. 27. (a) Find a spanning tree of ...
 7.25: (a) Find all the spanning trees of the network shown in Fig. 28(a)....
 7.26: (a) Find all the spanning trees of the network shown in Fig. 29(a)....
 7.27: (a) How many different spanning trees does the network shown in Fig...
 7.28: (a) How many different spanning trees does the network shown in Fig...
 7.29: Consider the network shown in Fig. 32. (a) How many different spann...
 7.30: Consider the network shown in Fig. 33. (a) How many different spann...
 7.31: The 3 by 4 grid shown in Fig. 34 represents a network of streets (3...
 7.32: The 4 by 5 grid shown in Fig. 35 represents a network of streets (4...
 7.33: Find the MST of the network shown in Fig. 36 using Kruskals algorit...
 7.34: Find the MST of the network shown in Fig. 37 using Kruskals algorit...
 7.35: Find the MST of the network shown in Fig. 38 using Kruskals algorit...
 7.36: Find the MST of the network shown in Fig. 39 using Kruskals algorit...
 7.37: Find the MaxST of the network shown in Fig. 36 using Kruskals algor...
 7.38: Find the MaxST of the network shown in Fig. 37 using Kruskals algor...
 7.39: Find the MaxST of the network shown in Fig. 38 using Kruskals algor...
 7.40: Find the MaxST of the network shown in Fig. 39 using Kruskals algor...
 7.41: . The mileage chart in Fig. 40 shows the distances between Atlanta,...
 7.42: The mileage chart in Fig. 41 shows the distances between Boston, Da...
 7.43: Figure 42(a) shows a network of roads connecting cities A through G...
 7.44: . Consider a network with M edges. Let k denote the number of bridg...
 7.45: . (a) Let G be a tree with N vertices. Find the sum of the degrees ...
 7.46: Explain why in a network with no loops or multiple edges, the maxim...
 7.47: This exercise refers to weighted networks where all the weights in ...
 7.48: This exercise refers to weighted networks where the weights in the ...
 7.49: . Suppose that in a weighted network there is just one edge (call i...
 7.50: Suppose that in a weighted network there is just one edge (call it ...
 7.51: Suppose G is a disconnected graph with N vertices, M edges, and no ...
 7.52: Suppose G is a disconnected graph with no circuits. Let N denote th...
 7.53: Cayleys theorem. Cayleys theorem says that the number of spanning t...
 7.54: Show that if a tree has a vertex of degree K, then there are at lea...
 7.55: A bipartite graph is a graph with the property that the vertices of...
 7.56: Suppose that there is an edge in a network that must be included in...
Solutions for Chapter 7: The Mathematics of Networks
Full solutions for Excursions in Modern Mathematics  8th Edition
ISBN: 9781292022048
Solutions for Chapter 7: The Mathematics of Networks
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Excursions in Modern Mathematics, edition: 8. Chapter 7: The Mathematics of Networks includes 56 full stepbystep solutions. Since 56 problems in chapter 7: The Mathematics of Networks have been answered, more than 13560 students have viewed full stepbystep solutions from this chapter. Excursions in Modern Mathematics was written by and is associated to the ISBN: 9781292022048.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

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

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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)·(b  Ax) = o.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

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 one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

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

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).