- 14.1.1: Show that each of the graphs in Figure 14.13 is planar by drawing i...
- 14.1.2: . Determine the boundaries of the regions of the plane graph G in F...
- 14.1.3: Let G be a planar graph and let G1 and G2 be two plane graphs obtai...
- 14.1.4: Let G be a connected 3-regular plane graph of order 8. How many reg...
- 14.1.5: Let G be a connected plane graph of order 8 having five regions. Ho...
- 14.1.6: The degrees of the vertices of a certain graph G are 3, 4, 4, 4, 5,...
- 14.1.7: The degrees of the vertices of a certain graph G are 2, 2, 2, 3, 5,...
- 14.1.8: A graph G of order n 3 has size m = 3n 6. Indicate which of the fol...
- 14.1.9: A planar graph G of order n 3 has size m < 3n6. Let u and v be two ...
- 14.1.10: (a) Give an example of a planar graph where no vertex of G has degr...
- 14.1.11: A graph G has a vertex of degree 4 and all other vertices of G have...
- 14.1.12: A certain graph G of order 6 has the following properties: (a) The ...
- 14.1.13: State and solve the Five Houses and Two Utilities Problem.
- 14.1.14: Determine all complete graphs Kn that are planar.
- 14.1.15: Determine all complete bipartite graphs Ks,t that are planar, where...
- 14.1.16: A graph G is a subdivision of a graph G having order n and size m. ...
- 14.1.17: Let U be a subset of the vertex set of a graph G and suppose that H...
- 14.1.18: Show that the Petersen graph (see Figure 14.15) is nonplanar by Kur...
- 14.1.19: Determine, with explanation, which graphs in Figure 14.16 are planar.
- 14.1.20: (a) Prove that if G is a planar graph of order n 3 and size m witho...
- 14.1.21: Let G be a plane graph of order n 5 and size m. (a) Prove that if t...
- 14.1.22: Determine all integers n 3 such that Cn is planar.
Solutions for Chapter 14.1: Planar Graphs
Full solutions for Discrete Mathematics | 1st Edition
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
Remove row i and column j; multiply the determinant by (-I)i + j •
Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
A symmetric matrix with eigenvalues of both signs (+ and - ).
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.
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. 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.
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
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.
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.
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