 Chapter 14.1: A finite set of points connected by line segments or curves is call...
 Chapter 14.2: Two graphs that have the same number of vertices connected to each ...
 Chapter 14.3: The number of edges at a vertex is called the __________ of the ver...
 Chapter 14.4: If there is at least one edge connecting two vertices in a graph, t...
 Chapter 14.5: If an edge is removed from a connected graph and leaves behind a di...
 Chapter 14.6: True or False: A graph can be drawn in many equivalent ways. ______
 Chapter 14.7: True or False: An edge can be a part of a path only once. ______
 Chapter 14.8: True or False: Every circuit is a path. ______
 Chapter 14.9: True or False: Every path is a circuit. ______
 Chapter 14.10: True or False: Eulers Theorem provides a procedure for finding Eule...
 Chapter 14.11: Draw a graph that models the bordering relationships among the stat...
 Chapter 14.12: Draw a graph that models the connecting relationships in the floor ...
 Chapter 14.13: In 1315, a graph is given. a. Determine whether the graph has an Eu...
 Chapter 14.14: In 1315, a graph is given. a. Determine whether the graph has an Eu...
 Chapter 14.15: In 1315, a graph is given. a. Determine whether the graph has an Eu...
 Chapter 14.16: Use Fleurys Algorithm to find an Euler path.
 Chapter 14.17: Use Fleurys Algorithm to find an Euler circuit.
 Chapter 14.18: Refer to Exercise 10. a. Use your graph to determine if the city re...
 Chapter 14.19: Refer to Exercise 11. Use your graph to determine if it is possible...
 Chapter 14.20: Refer to Exercise 12. a. Use your graph to determine if it is possi...
 Chapter 14.21: A security guard needs to walk the streets of the neighborhood in t...
 Chapter 14.22: In 2223, use the graph shown. A B C D EFind a Hamilton circuit that...
 Chapter 14.23: In 2223, use the graph shown. A B C D EFind a Hamilton circuit that...
 Chapter 14.24: For each graph in 2427, a. Determine if the graph must have Hamilto...
 Chapter 14.25: For each graph in 2427, a. Determine if the graph must have Hamilto...
 Chapter 14.26: For each graph in 2427, a. Determine if the graph must have Hamilto...
 Chapter 14.27: For each graph in 2427, a. Determine if the graph must have Hamilto...
 Chapter 14.28: Find the total weight of each of the six possible Hamilton circuits...
 Chapter 14.29: Use your answers from Exercise 28 and the Brute Force Method to fin...
 Chapter 14.30: Use the Nearest Neighbor Method, with starting vertex A, to find an...
 Chapter 14.31: Use the Nearest Neighbor Method to find a Hamilton circuit that beg...
 Chapter 14.32: The table shows the oneway airfares between cities. Use this infor...
 Chapter 14.33: The table shows the oneway airfares between cities. Use this infor...
 Chapter 14.34: In 3436, determine whether each graph is a tree. If the graph is no...
 Chapter 14.35: In 3436, determine whether each graph is a tree. If the graph is no...
 Chapter 14.36: In 3436, determine whether each graph is a tree. If the graph is no...
 Chapter 14.37: In 3738, find a spanning tree for each connected graph.
 Chapter 14.38: In 3738, find a spanning tree for each connected graph.
 Chapter 14.39: In 3940, use Kruskals Algorithm to find the minimum spanning tree f...
 Chapter 14.40: In 3940, use Kruskals Algorithm to find the minimum spanning tree f...
 Chapter 14.41: A fiberoptic cable system is to be installed along highways connec...
Solutions for Chapter Chapter 14: Graph Theory
Full solutions for Thinking Mathematically  6th Edition
ISBN: 9780321867322
Solutions for Chapter Chapter 14: Graph Theory
Get Full SolutionsChapter Chapter 14: Graph Theory includes 41 full stepbystep solutions. This textbook survival guide was created for the textbook: Thinking Mathematically, edition: 6. Since 41 problems in chapter Chapter 14: Graph Theory have been answered, more than 65787 students have viewed full stepbystep solutions from this chapter. Thinking Mathematically was written by and is associated to the ISBN: 9780321867322. This expansive textbook survival guide covers the following chapters and their solutions.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

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

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(DÂ» O.

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.

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

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.