 7.2.1: Rework Example 3 of Chapter 2 using the theorem on Euler paths. Her...
 7.2.2: a. Add a single arc to the graph of Exercise 1 so that there is an ...
 7.2.3: For Exercises 312, determine whether the given graph has an Euler p...
 7.2.4: For Exercises 312, determine whether the given graph has an Euler p...
 7.2.5: For Exercises 312, determine whether the given graph has an Euler p...
 7.2.6: For Exercises 312, determine whether the given graph has an Euler p...
 7.2.7: For Exercises 312, determine whether the given graph has an Euler p...
 7.2.8: For Exercises 312, determine whether the given graph has an Euler p...
 7.2.9: For Exercises 312, determine whether the given graph has an Euler p...
 7.2.10: For Exercises 312, determine whether the given graph has an Euler p...
 7.2.11: For Exercises 312, determine whether the given graph has an Euler p...
 7.2.12: For Exercises 312, determine whether the given graph has an Euler p...
 7.2.13: Draw the adjacency matrix for the graph of Exercise 3. In applying ...
 7.2.14: Draw the adjacency matrix for the graph of Exercise 5. In applying ...
 7.2.15: Draw the adjacency matrix for the graph of Exercise 7. In applying ...
 7.2.16: Draw the adjacency matrix for the graph of Exercise 9. In applying ...
 7.2.17: Describe two conditions on a connected directed graph, either of wh...
 7.2.18: Determine whether this graph has an Euler path. If so, list the nod...
 7.2.19: Determine whether this graph has an Euler path. If so, list the nod...
 7.2.20: Determine whether this graph has an Euler path. If so, list the nod...
 7.2.21: For Exercises 2128, decide by trial and error whether Hamiltonian c...
 7.2.22: For Exercises 2128, decide by trial and error whether Hamiltonian c...
 7.2.23: For Exercises 2128, decide by trial and error whether Hamiltonian c...
 7.2.24: For Exercises 2128, decide by trial and error whether Hamiltonian c...
 7.2.25: For Exercises 2128, decide by trial and error whether Hamiltonian c...
 7.2.26: For Exercises 2128, decide by trial and error whether Hamiltonian c...
 7.2.27: For Exercises 2128, decide by trial and error whether Hamiltonian c...
 7.2.28: For Exercises 2128, decide by trial and error whether Hamiltonian c...
 7.2.29: Prove that any graph with a Hamiltonian circuit is connected.
 7.2.30: Find an example of an unconnected graph that has an Euler path. (Hi...
 7.2.31: Consider a simple, complete graph with n nodes. Testing for a Hamil...
 7.2.32: Is it possible to walk in and out of each room in the house shown i...
 7.2.33: Recall that Kn denotes the simple, complete graph of order n. a. Fo...
 7.2.34: Recall that Km, n denotes a bipartite, complete graph with m + n no...
 7.2.35: Prove that a Hamiltonian circuit always exists in a connected graph...
 7.2.36: Consider a connected graph with 2n odd vertices, n 2. By the theore...
 7.2.37: Ores theorem (Oystein Ore, 1960) states that a Hamiltonian circuit ...
 7.2.38: Ores theorem (Exercise 37) gives a sufficient condition for a Hamil...
Solutions for Chapter 7.2: Euler Path and Hamiltonian Circuit
Full solutions for Mathematical Structures for Computer Science  7th Edition
ISBN: 9781429215107
Solutions for Chapter 7.2: Euler Path and Hamiltonian Circuit
Get Full SolutionsThis textbook survival guide was created for the textbook: Mathematical Structures for Computer Science, edition: 7. Mathematical Structures for Computer Science was written by and is associated to the ISBN: 9781429215107. This expansive textbook survival guide covers the following chapters and their solutions. Since 38 problems in chapter 7.2: Euler Path and Hamiltonian Circuit have been answered, more than 20004 students have viewed full stepbystep solutions from this chapter. Chapter 7.2: Euler Path and Hamiltonian Circuit includes 38 full stepbystep solutions.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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

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.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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

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.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

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

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.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

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.