 10.2.1: In the graph below, determine whether the following walks are trail...
 10.2.2: In the graph below, determine whether the following walks are trail...
 10.2.3: Let G be the graph v1 v2 e2 e1 and consider the walk v1e1v2e2v1. a....
 10.2.4: Consider the following graph.
 10.2.5: Consider the following graph. e2 e1 e5 e3 e4 a c b a. How many path...
 10.2.6: An edge whose removal disconnects the graph of which it is a part i...
 10.2.7: Given any positive integer n, (a) find a connected graph with n edg...
 10.2.8: Find the number of connected components for each of the following g...
 10.2.9: Each of (a)(c) describes a graph. In each case answer yes, no, or n...
 10.2.10: The solution for Example 10.2.5 shows a graph for which every verte...
 10.2.11: Is it possible for a citizen of Knigsberg to make a tour of the cit...
 10.2.12: Determine which of the graphs in 1217 have Euler circuits. If the g...
 10.2.13: Determine which of the graphs in 1217 have Euler circuits. If the g...
 10.2.14: Determine which of the graphs in 1217 have Euler circuits. If the g...
 10.2.15: Determine which of the graphs in 1217 have Euler circuits. If the g...
 10.2.16: Determine which of the graphs in 1217 have Euler circuits. If the g...
 10.2.17: Determine which of the graphs in 1217 have Euler circuits. If the g...
 10.2.18: Is it possible to take a walk around the city whose map is shown be...
 10.2.19: For each of the graphs in 1921, determine whether there is an Euler...
 10.2.20: For each of the graphs in 1921, determine whether there is an Euler...
 10.2.21: For each of the graphs in 1921, determine whether there is an Euler...
 10.2.22: The following is a floor plan of a house. Is it possible to enter t...
 10.2.23: Find Hamiltonian circuits for each of the graphs in 23 and 25
 10.2.24: Find Hamiltonian circuits for each of the graphs in 23 and 26
 10.2.25: Show that none of the graphs in 2527 has a Hamiltonian circuit.
 10.2.26: Show that none of the graphs in 2527 has a Hamiltonian circuit.
 10.2.27: Show that none of the graphs in 2527 has a Hamiltonian circuit.
 10.2.28: In 2831 find Hamiltonian circuits for those graphs that have them. ...
 10.2.29: In 2831 find Hamiltonian circuits for those graphs that have them. ...
 10.2.30: In 2831 find Hamiltonian circuits for those graphs that have them. ...
 10.2.31: In 2831 find Hamiltonian circuits for those graphs that have them. ...
 10.2.32: Give two examples of graphs that have Euler circuits but not Hamilt...
 10.2.33: Give two examples of graphs that have Hamiltonian circuits but not ...
 10.2.34: Give two examples of graphs that have circuits that are both Euler ...
 10.2.35: Give two examples of graphs that have Euler circuits and Hamiltonia...
 10.2.36: A traveler in Europe wants to visit each of the cities shown on the...
 10.2.37: a. Prove that if a walk in a graph contains a repeated edge, then t...
 10.2.38: Prove Lemma 10.2.1(a): If G is a connected graph, then any two dist...
 10.2.39: Prove Lemma 10.2.1(b): If vertices v and w are part of a circuit in...
 10.2.40: Draw a picture to illustrate Lemma 10.2.1(c): If a graph G is conne...
 10.2.41: Prove that if there is a trail in a graph G from a vertex v to a ve...
 10.2.42: If a graph contains a circuit that starts and ends at a vertex v, d...
 10.2.43: Prove that if there is a circuit in a graph that starts and ends at...
 10.2.44: Let G be a connected graph, and let C be any circuit in G that does...
 10.2.45: Prove that any graph with an Euler circuit is connected.
 10.2.46: Prove Corollary 10.2.5
 10.2.47: For what values of n does the complete graph Kn with n vertices hav...
 10.2.48: For what values of m and n does the complete bipartite graph on (m,...
 10.2.49: What is the maximum number of edges a simple disconnected graph wit...
 10.2.50: Show that a graph is bipartite if, and only if, it does not have a ...
Solutions for Chapter 10.2: Trails, Paths, and Circuits
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 10.2: Trails, Paths, and Circuits
Get Full SolutionsChapter 10.2: Trails, Paths, and Circuits includes 50 full stepbystep solutions. Since 50 problems in chapter 10.2: Trails, Paths, and Circuits have been answered, more than 28128 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4th. Discrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326.

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.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

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

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.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

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

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

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.