 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 3regular 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
ISBN: 9781577667308
Solutions for Chapter 14.1: Planar Graphs
Get Full SolutionsThis textbook survival guide was created for the textbook: Discrete Mathematics, edition: 1. This expansive textbook survival guide covers the following chapters and their solutions. Discrete Mathematics was written by Patricia and is associated to the ISBN: 9781577667308. Chapter 14.1: Planar Graphs includes 22 full stepbystep solutions. Since 22 problems in chapter 14.1: Planar Graphs have been answered, more than 4526 students have viewed full stepbystep solutions from this chapter.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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

Companion matrix.
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 nl  An).

Cyclic shift
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.

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.

Distributive Law
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 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Indefinite matrix.
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 AI.
Square matrix with AI A = I and AAl = 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 B1 AI and (AI)T. Cofactor formula (Al)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, ... , 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.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. 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.

Schwarz inequality
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.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Vector addition.
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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