 14.2.14.1.47: A path that passes through each edge of a graph exactly one time is...
 14.2.14.1.48: A circuit that passes through each edge of a graph exactly one time...
 14.2.14.1.49: A connected graph has at least one Euler path that is also an Euler...
 14.2.14.1.50: A connected graph has at least one Euler path, but no Euler circuit...
 14.2.14.1.51: A connected graph has neither an Euler path nor an Euler circuit, i...
 14.2.14.1.52: If a connected graph has exactly two odd vertices, A and B, then ea...
 14.2.14.1.53: For Exercises 710, use the following graph. A B D E C.Determine an ...
 14.2.14.1.54: For Exercises 710, use the following graph. A B D E C.Determine an ...
 14.2.14.1.55: For Exercises 710, use the following graph. A B D E C.Is it possibl...
 14.2.14.1.56: For Exercises 710, use the following graph. A B D E C.Is it possibl...
 14.2.14.1.57: For Exercises 1114, use the following graph. A E B D C.Determine an...
 14.2.14.1.58: For Exercises 1114, use the following graph. A E B D C.Determine an...
 14.2.14.1.59: For Exercises 1114, use the following graph. A E B D C.Is it possib...
 14.2.14.1.60: For Exercises 1114, use the following graph. A E B D C.Is it possib...
 14.2.14.1.61: For Exercises 1520, use the following graph. E D F C A B.Determine ...
 14.2.14.1.62: For Exercises 1520, use the following graph. E D F C A B.Determine ...
 14.2.14.1.63: For Exercises 1520, use the following graph. E D F C A B.Determine ...
 14.2.14.1.64: For Exercises 1520, use the following graph. E D F C A B.Determine ...
 14.2.14.1.65: For Exercises 1520, use the following graph. E D F C A B.Determine ...
 14.2.14.1.66: For Exercises 1520, use the following graph. E D F C A B.Determine ...
 14.2.14.1.67: Imagine a very large connected graph that has 400 even vertices and...
 14.2.14.1.68: Imagine a very large connected graph that has two odd vertices and ...
 14.2.14.1.69: Imagine a very large connected graph that has 400 odd vertices and ...
 14.2.14.1.70: Imagine a very large connected graph that has 200 odd vertices and ...
 14.2.14.1.71: Revisiting the Knigsberg Bridge Exercises 25 and 26, suppose that t...
 14.2.14.1.72: Revisiting the Knigsberg Bridge Exercises 25 and 26, suppose that t...
 14.2.14.1.73: Other Navy Regions In Exercises 27 and 28, the maps of states that ...
 14.2.14.1.74: Other Navy Regions In Exercises 27 and 28, the maps of states that ...
 14.2.14.1.75: Areas of the World In Exercises 2932 use each map shown. a) Represe...
 14.2.14.1.76: Areas of the World In Exercises 2932 use each map shown. a) Represe...
 14.2.14.1.77: Areas of the World In Exercises 2932 use each map shown. a) Represe...
 14.2.14.1.78: Areas of the World In Exercises 2932 use each map shown. a) Represe...
 14.2.14.1.79: Locking Doors Joe Mays, the custodian at the Pullen Academy (see Ex...
 14.2.14.1.80: Locking Doors Joe Mays, the custodian at the Pullen Academy (see Ex...
 14.2.14.1.81: Locking Doors Joe Mays, the custodian at the Pullen Academy (see Ex...
 14.2.14.1.82: Locking Doors Joe Mays, the custodian at the Pullen Academy (see Ex...
 14.2.14.1.83: Crime Stopper Routes The Country Oaks crime stopper organization (s...
 14.2.14.1.84: Crime Stopper Routes The Country Oaks crime stopper organization (s...
 14.2.14.1.85: In Exercises 39 42, use Fleurys algorithm to determine an Euler path.
 14.2.14.1.86: In Exercises 39 42, use Fleurys algorithm to determine an Euler path.
 14.2.14.1.87: In Exercises 39 42, use Fleurys algorithm to determine an Euler path.
 14.2.14.1.88: In Exercises 39 42, use Fleurys algorithm to determine an Euler path.
 14.2.14.1.89: In Exercises 43 48, use Fleurys algorithm to determine an Euler cir...
 14.2.14.1.90: In Exercises 43 48, use Fleurys algorithm to determine an Euler cir...
 14.2.14.1.91: In Exercises 43 48, use Fleurys algorithm to determine an Euler cir...
 14.2.14.1.92: In Exercises 43 48, use Fleurys algorithm to determine an Euler cir...
 14.2.14.1.93: In Exercises 43 48, use Fleurys algorithm to determine an Euler cir...
 14.2.14.1.94: In Exercises 43 48, use Fleurys algorithm to determine an Euler cir...
 14.2.14.1.95: Determine an Euler path through the states in the Navy Region South...
 14.2.14.1.96: Determine an Euler path for the kindergarten building at the Pullen...
 14.2.14.1.97: Determine an Euler circuit for the Country Oaks crime stopper group...
 14.2.14.1.98: Determine an Euler circuit for the Country Oaks crime stopper group...
 14.2.14.1.99: Determine an Euler circuit for the Country Oaks crime stopper group...
 14.2.14.1.100: Determine an Euler circuit for the Country Oaks crime stopper group...
 14.2.14.1.101: Consider a map of the contiguous United States. Imagine a graph wit...
 14.2.14.1.102: Attempt to draw a graph with an Euler circuit that has a bridge. Wh...
 14.2.14.1.103: a) Draw a graph with one vertex that has both an Euler path and an ...
 14.2.14.1.104: Write a paper on the history and development of the branch of mathe...
Solutions for Chapter 14.2: Graph Theory
Full solutions for A Survey of Mathematics with Applications  9th Edition
ISBN: 9780321759665
Solutions for Chapter 14.2: Graph Theory
Get Full SolutionsA Survey of Mathematics with Applications was written by and is associated to the ISBN: 9780321759665. This textbook survival guide was created for the textbook: A Survey of Mathematics with Applications, edition: 9. Chapter 14.2: Graph Theory includes 58 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 58 problems in chapter 14.2: Graph Theory have been answered, more than 75043 students have viewed full stepbystep solutions from this chapter.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

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

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

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

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.

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.

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

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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

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.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.