the transitive closure of a relation of
your choice on a set with at least 20
elements. Either use a relation that
corresponds to direct links in a
particular transportation or
communications network or use a
randomly generated relation. 6.
Compute the nu
element of the lattice has a
complement. 46. Give an example of a
finite lattice where at least one
element has more than one
complement and at least one element
has no complement. 47. Show that the
lattice (P (S), ) where P (S) is the
power set of a fini
is an edge of a simple graph associated
to cfw_u, v, we can also say, without
possible confusion, that cfw_u, v is an
edge of the graph. A computer network
may contain multiple links between
data centers, as shown in Figure 2. To
model such networks we ne
that link particular types of
information. Here, we will describe
how to model the World Wide Web
using a graph. We will also describe
how to use a graph to model the
citations in different types of
documents. EXAMPLE 5 The Web
Graph The World Wide Web ca
special properties? 32. Which
statements must be executed before S6
is executed in the program in Example
8? (Use the precedence graph in Figure
10.) 33. Construct a precedence graph
for the following program: S1: x := 0 S2:
x := x + 1 S3: y := 2 S4: z :=
flight from Washington to Miami, with
a) an edge between vertices
representing cities that have a flight
between them (in either direction). b)
an edge between vertices representing
cities for each flight that operates
between them (in either direction).
If the graph is directed, are multiple
directed edges present? Are loops
present? Answering such questions
helps us understand graphs. It is less
important to remember the particular
terminology used. Graph Models
Graphs are used in a wide variety of
mode
Graphs and Graph Models We begin
with the definition of a graph.
DEFINITION 1 A graph G = (V , E)
consists of V , a nonempty set of
vertices (or nodes) and E, a set of
edges. Each edge has either one or two
vertices associated with it, called its
endpoint
We will show how graphs can be used
to model roadmaps and the
assignment of jobs to employees of an
organization. Using graph models, we
can determine whether it is possible to
walk down all the streets in a city
without going down a street twice, and
we
once). Such a directed graph model is
presented in Figure 13. We see that
Team 1 is undefeated in this
tournament, and Team 3 is winless.
EXAMPLE 14 SingleEliminationTournaments A
tournament where each contestant is
eliminated after one loss is called a
collaborated extensively and who took
care of many of his worldly needs.
Erdos offered rewards, ranging from
$10 to $10,000, for the solution of
problems that he found particularly
interesting, with the size of the reward
depending on the difficulty of th
are n + 1 people such that none of
these people is a descendant of any of
the other n people. [Hint: Use Exercise
32.] Suppose that (S, ) is a well-founded
partially ordered set. The principle of
well-founded induction states that P
(x) is true for all x
graph is a simple graph because no
loops or multiple edges are needed in
this model. The graph in Figure 11
models the ecosystem of a forest. We
see from this graph that squirrels and
raccoons compete but that crows and
shrews do not. EXAMPLE 12 Protein
I
Warshalls algorithm was discovered
independently by more than one
person? 4. Describe how equivalence
classes can be used to define the
rational numbers as classes of pairs of
integers and how the basic arithmetic
operations on rational numbers can be
def
that the following properties hold for
all elements x, y, and z of a lattice L. a) x
y = y x and x y = y x
(commutative laws) b) (x y) z = x
(y z) and (x y) z = x (y z)
(associative laws) c) x (x y) = x and
x (x y) = x (absorption laws) d) x
x = x and
message to many different e-mail
addresses? 27. Describe a graph model
that represents whether each person
at a party knows the name of each
other person at the party. Should the
edges be directed or undirected?
Should multiple edges be allowed?
Should lo
1 A Computer Network. graphs by
using points to represent vertices and
line segments, possibly curved, to
represent edges, where the endpoints
of a line segment representing an edge
are the points representing the
endpoints of the edge. When we draw a
gra
643 San Francisco Los Angeles Denver
Detroit Chicago New York Washington
FIGURE 4 A Communications Network
with One-Way Communications Links.
need to include edges that connect a
vertex to itself. Such edges are called
loops, and sometimes we may even
hav
the vertex representing the squirrel in
the niche overlap graph in Figure 11 in
Section 10.1 is four, because the
squirrel competes with four other
species: the crow, the opossum, the
raccoon, and the woodpecker. In this
niche overlap graph, the mouse is
represented by a vertex. An undirected
edge is used to connect two people
when these people know each other,
when we are concerned only with
acquaintanceship, or whether they are
friends. No multiple edges and usually
no loops are used. (If we want to
inc
influenced, but she can influence
Brian, Fred, and Linda. Also, Yvonne
and Brian can influence each other.
EXAMPLE 3 Collaboration Graphs A
collaboration graph is used to model
social networks where two people are
related by working together in a
particu
we denote by N (A) the set of all
vertices in G that are adjacent to at
least one vertex in A. So, N (A) = vA N
(v). To keep track of how many edges
are incident to a vertex, we make the
following definition. DEFINITION 3 The
degree of a vertex in an undi
papers in mathematics was found in
2004 to have more than 400,000
vertices and 675,000 edges, and these
numbers have grown considerably
since then. We will have more to say
about this graph in Section 10.4.
Collaboration graphs have also been
used in spor
functions and the functionality of
newly discovered proteins. Because
there are thousands of different
proteins in a typical cell, the protein
interaction graph of a cell is extremely
large and complex. For example, yeast
cells have more than 6,000 protei
argument in Example 12 in Section
1.8.] Computer Projects Write
programs with these input and output.
1. Given the matrix representing a
relation on a finite set, determine
whether the relation is reflexive
and/or irreflexive. 2. Given the matrix
represen
at the telephone number to which the
call was made. We need directed edges
because the direction in which the call
is made matters. We need multiple
directed edges because we want to
represent each call made from a
particular telephone number to a
second
graphs to represent relations.
DEFINITION 2 A directed graph (or
digraph) (V , E) consists of a nonempty
set of vertices V and a set of directed
edges (or arcs) E. Each directed edge is
associated with an ordered pair of
vertices. The directed edge associ