 10.4.1E: For each pair of graphs G and G$ in 1–5, determine whether G and G’...
 10.4.2E: For each pair of graphs G and G$ in 1–5, determine whether G and G’...
 10.4.3E: For each pair of graphs G and G$ in 1–5, determine whether G and G’...
 10.4.4E: For each pair of graphs G and G$ in 1–5, determine whether G and G’...
 10.4.5E: For each pair of graphs G and G$ in 1–5, determine whether G and G’...
 10.4.6E: For each pair of simple graphs G and G’ in 6–13, determine whether ...
 10.4.7E: For each pair of simple graphs G and G’ in 6–13, determine whether ...
 10.4.8E: For each pair of simple graphs G and G’ in 6–13, determine whether ...
 10.4.9E: For each pair of simple graphs G and G’ in 6–13, determine whether ...
 10.4.10E: For each pair of simple graphs G and G’ in 6–13, determine whether ...
 10.4.11E: For each pair of simple graphs G and G’ in 6–13, determine whether ...
 10.4.12E: For each pair of simple graphs G and G’ in 6–13, determine whether ...
 10.4.13E: For each pair of simple graphs G and G’ in 6–13, determine whether ...
 10.4.14E: Draw all nonisomorphic simple graphs with three vertices.
 10.4.15E: Draw all nonisomorphic simple graphs with four vertices.
 10.4.16E: Draw all nonisomorphic graphs with three vertices and no more than ...
 10.4.17E: Draw all nonisomorphic graphs with four vertices and no more than t...
 10.4.18E: Draw all nonisomorphic graphs with four vertices and three edges.
 10.4.19E: Draw all nonisomorphic graphs with six vertices, all having degree 2.
 10.4.20E: Draw four nonisomorphic graphs with six vertices, two of degree 4 a...
 10.4.21E: Prove that each of the properties in 21–29 is an invariant for grap...
 10.4.22E: Prove that each of the properties in 21–29 is an invariant for grap...
 10.4.23E: Prove that each of the properties in 21–29 is an invariant for grap...
 10.4.24E: Prove that each of the properties in 21–29 is an invariant for grap...
 10.4.25E: Prove that each of the properties in 21–29 is an invariant for grap...
 10.4.26E: Prove that each of the properties in 21–29 is an invariant for grap...
 10.4.27E: Prove that each of the properties in 21–29 is an invariant for grap...
 10.4.28E: Prove that each of the properties in 21–29 is an invariant for grap...
 10.4.29E: Prove that each of the properties in 21–29 is an invariant for grap...
 10.4.30E: Show that the following two graphs are not isomorphic by supposing ...
Solutions for Chapter 10.4: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 10.4
Get Full SolutionsSince 30 problems in chapter 10.4 have been answered, more than 23665 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Discrete Mathematics with Applications was written by Sieva Kozinsky and is associated to the ISBN: 9780495391326. Chapter 10.4 includes 30 full stepbystep solutions. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4th.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

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.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

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

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.

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].

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

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

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

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

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