 10.5.23E: A connected graph has nine vertices and twelve edges. Does it have ...
 10.5.1E: Read the tree in Example from left to right to answer the following...
 10.5.2E: Draw trees to show the derivations of the following sentences from ...
 10.5.3E: What is the total degree of a tree with n vertices? Why?
 10.5.4E: Let G be the graph of a hydrocarbon molecule with the maximum numbe...
 10.5.5E: Extend the argument given in the proof of Lemma to show that a tree...
 10.5.6E: If graphs are allowed to have an infinite number of vertices and ed...
 10.5.7E: Find all terminal vertices and all internal vertices for the follow...
 10.5.8E: In each, either draw a graph with the given specifications or expla...
 10.5.9E: In each, either draw a graph with the given specifications or expla...
 10.5.10E: In each, either draw a graph with the given specifications or expla...
 10.5.11E: In each, either draw a graph with the given specifications or expla...
 10.5.12E: In each, either draw a graph with the given specifications or expla...
 10.5.13E: In each, either draw a graph with the given specifications or expla...
 10.5.14E: In each, either draw a graph with the given specifications or expla...
 10.5.15E: In each, either draw a graph with the given specifications or expla...
 10.5.16E: In each, either draw a graph with the given specifications or expla...
 10.5.17E: In each, either draw a graph with the given specifications or expla...
 10.5.18E: In each, either draw a graph with the given specifications or expla...
 10.5.19E: In each, either draw a graph with the given specifications or expla...
 10.5.20E: In each, either draw a graph with the given specifications or expla...
 10.5.21E: In each, either draw a graph with the given specifications or expla...
 10.5.22E: A connected graph has twelve vertices and eleven edges. Does it hav...
 10.5.24E: Suppose that v is a vertex of degree 1 in a connected graph G and t...
 10.5.25E: A graph has eight vertices and six edges. Is it connected? Why?
 10.5.26E: If a graph has n vertices and n – 2 or fewer edges, can it be conne...
 10.5.27E: A circuitfree graph has ten vertices and nine edges. Is it connect...
 10.5.28E: Is a circuitfree graph with n vertices and at least n – 1 edges co...
 10.5.29E: Prove that every nontrivial tree has at least two vertices of degre...
 10.5.30E: Find all nonisomorphic trees with five vertices.
 10.5.31E: a. Prove that the following is an invariant for graph isomorphism: ...
Solutions for Chapter 10.5: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 10.5
Get Full SolutionsChapter 10.5 includes 31 full stepbystep solutions. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4. Since 31 problems in chapter 10.5 have been answered, more than 57229 students have viewed full stepbystep solutions from this chapter. Discrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326. This expansive textbook survival guide covers the following chapters and their solutions.

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).

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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].

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

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.

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

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.

Iterative method.
A sequence of steps intended to approach the desired solution.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

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

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.