 12.4.1: Figure 12.59 shows a diagram of an art exhibit consisting of seven ...
 12.4.2: Determine which of the graphs in Figure 12.60 are Hamiltonian.
 12.4.3: (a) What does Theorem 12.59 say about Q3? (b) Is Q3 Hamiltonian?
 12.4.4: (a) Show that there exists no graph G such that both G and G satisf...
 12.4.5: Show that if a connected graph G contains three vertices of degree ...
 12.4.6: Suppose that G is a graph of order n 3 such that deg v (n 1)/2 for ...
 12.4.7: The vertices of a Hamiltonian graph G of order n have degrees d1, d...
 12.4.8: Determine which of the graphs P3 P2, P3 P3, P3 P4 and P3 P5 are Ham...
 12.4.9: Determine whether the graph C3 C3 is Hamiltonian.
 12.4.10: Does there exist a 4regular graph G of order 6 containing two Hami...
 12.4.11: Let Ks,t be the complete bipartite graph where 2 s t. Prove that Ks...
 12.4.12: Prove or disprove: If u and v are two adjacent vertices in a graph ...
 12.4.13: A Hamiltonian graph G of order n has the property that for every ed...
 12.4.14: Show that if every vertex of a graph G of order n 4 has degree at l...
 12.4.15: Prove or disprove: If G is a graph of order n 3 such that deg v < n...
 12.4.16: Let S = {1, 2, 3, 4} and let V be the set of 2element and 3elemen...
Solutions for Chapter 12.4: Hamiltonian Graphs
Full solutions for Discrete Mathematics  1st Edition
ISBN: 9781577667308
Solutions for Chapter 12.4: Hamiltonian Graphs
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Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

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!

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

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.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

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.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

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 .

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

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

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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.

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.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

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

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.