 10.2.29E: Suppose that there are five young women and five young men on an is...
 10.2.35E: How many vertices and how many edges do these graphs have?a) Kn____...
 10.2.33E: For the graph G in Exercise I finda) the subgraph induced by the ve...
 10.2.1E: In Exercises 1–3 find the number of vertices, the number of edges, ...
 10.2.2E: In Exercises 1–3 find the number of vertices, the number of edges, ...
 10.2.4E: Find the sum of the degrees of the vertices of each graph in Exerci...
 10.2.3E: In Exercises 1–3 find the number of vertices, the number of edges, ...
 10.2.5E: Can a simple graph exist with 15 vertices each of degree five?
 10.2.6E: Show that the sum. over the set of people at a party, of the number...
 10.2.7E: In Exercises 7–9 determine the number of vertices and edges and fin...
 10.2.8E: In Exercises 7–9 determine the number of vertices and edges and fin...
 10.2.9E: In Exercises 7–9 determine the number of vertices and edges and fin...
 10.2.10E: For each of the graphs in Exercise 7–9 determine the sum of the in...
 10.2.11E: Construct the underlying undirected graph for the graph with direct...
 10.2.12E: What does the degree of a vertex represent in the acquaintanceship ...
 10.2.13E: What does the degree of a vertex represent in an academic collabora...
 10.2.14E: What does the degree of a vertex in the Hollywood graph represent? ...
 10.2.17E: What do the indegree and the outdegree of a vertex in a directed ...
 10.2.15E: What do the indegree and the outdegree of a vertex in a telephone...
 10.2.16E: What do the indegree and the outdegree of a vertex in the Web gra...
 10.2.19E: Use Exercise 18 to show that in a group of people, there must be tw...
 10.2.18E: Show that in a simple graph with at least two vertices there must b...
 10.2.20E: Draw these graphs.a) K7________________b) K1,8________________c) K4...
 10.2.21E: In Exercises 21–25 determine whether the graph is bipartite. You ma...
 10.2.23E: In Exercises 21–25 determine whether the graph is bipartite. You ma...
 10.2.22E: In Exercises 21–25 determine whether the graph is bipartite. You ma...
 10.2.28E: Suppose that a new company has five employees: Zamora. Agraharam. S...
 10.2.26E: For which values of n are these graphs bipartite?a) Kn_____________...
 10.2.27E: Suppose that there are four employees in the computer support group...
 10.2.25E: In Exercises 21–25 determine whether the graph is bipartite. You ma...
 10.2.24E: In Exercises 21–25 determine whether the graph is bipartite. You ma...
 10.2.30E: Suppose that there are five young women and six young men on an isl...
 10.2.31E: Suppose there is an integer k such that every man on a desert islan...
 10.2.32E: In this exercise we prove a theorem of 0ystein ?re. Suppose that G ...
 10.2.37E: Find the degree sequence of each of the following graphs.a) K4_____...
 10.2.38E: What is the degree sequence of the bipartite graph Km,n where m and...
 10.2.39E: What is the degree sequence of Kn, where n is a positive integer? E...
 10.2.36E: Find the degree sequences for each of the graphs in Exercises 21–25.
 10.2.34E: Let n be a positive integer. Show that a subgraph induced by a none...
 10.2.40E: How many edges does a graph have if its degree sequence is 4, 3, 3,...
 10.2.41E: How many edges does a graph have if its degree sequence is 5, 2, 2,...
 10.2.42E: Determine whether each of these sequences is graphic. For those tha...
 10.2.44E: Suppose that d1, d2,..., dn„ is a graphic sequence. Show that there...
 10.2.45E: Show that a sequence d1, d2...,dn of nonnegative integers in noninc...
 10.2.43E: Determine whether each of these sequences is graphic. For those tha...
 10.2.46E: Use Exercise 45 to construct a recursive algorithm for determining ...
 10.2.49E: How many subgraphs with at least one vertex does K3 have?
 10.2.50E: How many subgraphs with at least one vertex does W3 have?
 10.2.48E: How many subgraphs with at least one vertex does K2 have?
 10.2.47E: Show that every nonincreasing sequence of nonnegative integers with...
 10.2.51E: Draw all subgraphs of this graph.
 10.2.52E: Let G be a graph with v vertices and e edges. Let M be the maximum ...
 10.2.53E: For which values of n are these graphs regular?a) Kn_______________...
 10.2.54E: For which values of m and n is Km,n regular?
 10.2.55E: How many vertices does a regular graph of degree four with 10 edges...
 10.2.56E: In Exercises 56–58 find the union of the given pair of simple graph...
 10.2.61E: If the simple graph G has v vertices and e edges, how many edges do...
 10.2.57E: In Exercises 56–58 find the union of the given pair of simple graph...
 10.2.58E: In Exercises 56–58 find the union of the given pair of simple graph...
 10.2.59E: The complementary graph of a simple graph G has the same vertices a...
 10.2.60E: If G is a simple graph with 15 edges and has 13 edges, how many ver...
 10.2.62E: If the degree sequence of the simple graph G is 4. 3, 3, 2, 2, what...
 10.2.64E: Show that if G is a bipartite simple graph with v vertices and e ed...
 10.2.63E: If the degree sequence of the simple graph G is d1, d2 dn, what is ...
 10.2.66E: Describe an algorithm to decide whether a graph is bipartite based ...
 10.2.65E: Show that if G is a simple graph with n vertices, then the union of...
 10.2.70E: Show that if a bipartite graph G = (V. E) is nregular for some pos...
 10.2.68E: Show that (Gconv)conv = G whenever G is a directed graph.
 10.2.69E: Show that the graph G is its own converse if and only if the relati...
 10.2.67E: Draw the converse of each of the graphs in Exercises 7–9 in Section...
 10.2.71E: Draw the mesh network for interconnecting nine parallel processors.
 10.2.72E: In a variant of a mesh network for interconnecting?; = m2processors...
 10.2.73E: Show that every pair of processors in a mesh network of n = m2 proc...
Solutions for Chapter 10.2: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications  7th Edition
ISBN: 9780073383095
Solutions for Chapter 10.2
Get Full SolutionsSince 73 problems in chapter 10.2 have been answered, more than 219510 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095. Chapter 10.2 includes 73 full stepbystep solutions.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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.

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.

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

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

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.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

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

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Outer product uv T
= column times row = rank one matrix.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

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 ().

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.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.