 10.1.10E: For each undirected graph in Exercises 3–9 that is not simple, find...
 10.1.9E:
 10.1.11E: Let G be a simple graph. Show that the relation R on the set of ver...
 10.1.1E: Draw graph models, stating the type of graph (from Table 1) used, t...
 10.1.2E: What kind of graph (from Table 1) can be used to model a highway sy...
 10.1.3E:
 10.1.4E:
 10.1.6E:
 10.1.7E:
 10.1.5E:
 10.1.8E:
 10.1.14E: Use the niche overlap graph in Figure 11 to determine the species t...
 10.1.13E: The intersection graph of a collection of sets A1, A2, .... An is t...
 10.1.12E: Let G be an undirected graph with a loop at every vertex. Show that...
 10.1.16E: Draw the acquaintanceship graph that represents that Tom and Patric...
 10.1.15E: Construct a niche overlap graph for six species of birds, where the...
 10.1.17E: We can use a graph to represent whether two people were alive at th...
 10.1.18E: Who can influence Fred and whom can Fred influence in the influence...
 10.1.19E: Construct an influence graph for the board members of a company if ...
 10.1.21E: In a roundrobin tournament the Tigers beat the Blue Jays, the Tige...
 10.1.20E: Which other teams did Team 4 beat and which teams beat Team 4 in th...
 10.1.22E: Construct the call graph for a set of seven telephone numbers 555–0...
 10.1.25E: How can a graph that models email messages sent in a network be us...
 10.1.26E: How can a graph that models email messages sent in a network be us...
 10.1.24E: a) Explain how graphs can be used to model electronic mail messages...
 10.1.23E: Explain how the two telephone call graphs for calls made during the...
 10.1.28E: Describe a graph model that represents a subway system in a large c...
 10.1.29E: For each course at a university, there may be one or more other cou...
 10.1.27E: Describe a graph model that represents whether each person at a par...
 10.1.30E: Describe a graph model that represents the positive recommendations...
 10.1.31E: Describe a graph model that represents traditional marriages betwee...
 10.1.32E: Which statements must be executed before S(, is executed in the pro...
 10.1.34E: Describe a discrete structure based on a graph that can be used to ...
 10.1.33E: Construct a precedence graph for the following program:
 10.1.36E: Describe a graph model that can be used to represent all forms of e...
 10.1.35E: Describe a discrete structure based on a graph that can be used to ...
Solutions for Chapter 10.1: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications  7th Edition
ISBN: 9780073383095
Solutions for Chapter 10.1
Get Full SolutionsDiscrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095. Since 36 problems in chapter 10.1 have been answered, more than 221526 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 10.1 includes 36 full stepbystep solutions. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7.

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.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

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

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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).

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

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.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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

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).

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

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

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

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