Unit 3 progress check Discrete Mathematics
The following graph is connected.
FALSE
Identify the adjacency matrix for the given graph.
The matrix contains 1 as an entry at the (i, j)th position if there is an edge from vertex i to vertex j; otherwise, the entry is 0.
Select the directed graph for the given adjacency matrix.
The matrix contains 1 as an entry at the (i, j)th position if there is an edge from vertex i to vertex j; otherwise, the entry is 0.
Click and drag the steps to determine whether the given pair of graphs are isomorphic.
The second graph has a vertex of degree four, while the first graph does not. Hence, these graphs are not isomorphic.
Identify the following graphs: K1,8
The star graph K1,8 consists of a single vertex adjacent to all the other eight vertices. The graph of K1,8 is
Consider the graphs, Kn, Cn, Wn, Km,n, and Qn. What is the number of vertices and edges in the graph Km,n?
m + n vertices and mn edges
Consider the graphs, Kn, Cn, Wn, Km,n, and Qn. dentify the number of vertices and edges in the graph Cn.
n and n
Consider the graphs, Kn, Cn, Wn, Km,n, and Qn. The number of vertices and edges in Wn is (answer) n n + 1 2n n - 1 and (answer) 2 - n 2n n/2 n .
n+1 2n The number of vertices and edges in Wn is n + 1 and 2n .
Suppose that p and q are distinct primes. Use the principle of inclusion-exclusion to identify ϕ(pq)ϕ(pq) , the number of positive integers not exceeding pq that are relatively prime to pq.
(p - 1)(q - 1)
What is the general form of the particular solution guaranteed to exist by Theorem 6 of the linear nonhomogeneous recurrence relation an = 6an - 1 - 12an - 2 + 8an - 3 + F(n) if F(n) = n3(-2)n?
(p3n^3 + p2n^2 + p1n + p0)(-2)^n
How many connected components does the following graphs have?
1
Consider the directed multigraph. the vertices to their number of out-degree
c = 1 out-degree d= 1 out=degree a = 2 out-degree b = 4 out-degree
Consider the directed multigraph. What is the number of vertices for the given graph?
4
Consider the graphs, Kn, Cn, Wn, Km,n, and Qn. The graph Qn has 2n vertices and n2n − 1 edges.
TRUE
Let an be the number of ways to climb n stairs if a person climbing the stairs can take one stair or two stairs at a time Identify the initial condition for the recurrence relation in the previous question.
a0 = 1 and a1 = 1
Consider the nonhomogeneous linear recurrence relation an = 2an − 1 + 2n. Identify the solution of the given recurrence relation with a0 = 2.
an = (n + 2)2^n
Consider the nonhomogeneous linear recurrence relation an = 2an − 1 + 2n. Identify the set of all solutions of the given recurrence relation using the theorem given below. If {a(p)n} is a particular solution of the nonhomogeneous linear recurrence relation with constant coefficients an = c1an − 1 + c2an − 2 +· · ·+ckan − k + F(n), then every solution of the form {a(p)n + a(h)n}, where {a(h)n} is a solution of the associated homogeneous recurrence relation an = c1an − 1 + c2an − 2 +· · ·+ckan − k.
an = α(2)^n + n(2)^n
Find the number of paths between c and d in the following graph of length 2. There are 0 paths of length 2.
0
Find the number of elements in A1 ∪ A2 ∪ A3 if there are 100 elements in A1, 1000 in A2, and 10,000 in A3 if the sets are pairwise disjoint.
11100 The sets are pairwise disjoint, which implies A1 ∩ A2 = A2 ∩ A3 = A1 ∩ A3 = A1 ∩ A2 ∩ A3 = ∅. On substituting the values |A1 ∩ A2| = |A2 ∩ A3| = |A1 ∩ A3| = |A1 ∩ A2 ∩ A3| = 0 in |A1 ∪ A2 ∪ A3|, we get |A1 ∪ A2 ∪ A3| = |A1| + |A2| + |A3| - |A1 ∩ A2| - |A2 ∩ A3| - |A1 ∩ A3| + |A1 ∩ A2 ∩ A3| = |A1| + |A2| + |A3| |A1 ∪ A2 ∪ A3| = 100 + 1000 + 10,000 = 11,100
Find the number of elements in A1 ∪ A2 ∪ A3 if there are 120 elements in each set and if the sets are equal
120 We use the principle of inclusion-exclusion for three sets, i.e., |A1 ∪ A2 ∪ A3| = |A1| + |A2| + |A3| - |A1 ∩ A2| - |A1 ∩ A3| - |A2 ∩ A3| + |A1 ∩ A2 ∩ A3|. If all the sets are equal, then |A1| = |A2| = |A3| = |A1 ∩ A2| = |A1 ∩ A3| = |A2 ∩ A3| = |A1 ∩ A2 ∩ A3| = 120. |A1 ∪ A2 ∪ A3| = 120 + 120 + 120 - 120 - 120 - 120 + 120 = 120
Find the number of paths of length n between two different vertices in K4if n = 2. There are 2 Numeric ResponseEdit Unavailable.2correct.paths of length 2 between two vertices.
2
How many nonisomorphic simple graphs are there with n vertices for the following values of n? When n = 2, there are (answer) nonisomorphic simple graphs.
2 When n = 2, there are__ 2 __nonisomorphic simple graphs.
Identify the degree sequence of each of the following graphs. C4
2, 2, 2, 2
A sequence d1,d2,...,dn is called graphic if it is the degree sequence of a simple graph. Determine whether each of these sequences is graphic. (Check all that apply.) (You must provide an answer before moving to the next part.)
2, 2, 2, 2, 2, 2 3, 3, 2, 2, 2, 2 1, 1, 1, 1, 1, 1 5, 3, 3, 3, 3, 3
Find the number of paths of length n between two different vertices in K4if n = 4. There are 20 paths of length 4 between two vertices.
20
How many elements are in A1 ∪ A2 if there are 12 elements in A1, 18 elements in A2, and |A1 ∩ A2| = 1? The number of elements in A1 ∪ A2 is .
29 Using the principle of inclusion-exclusion, we have |A1 ∪ A2| = |A1| + |A2| - |A1 ∩ A2|. Therefore, |A1 ∪ A2| = 12 + 18 - 1 = 29.
Identify the degree sequence of each of the following graphs. K4
3, 3, 3, 3
Identify the degree sequence of each of the following graphs. Q3
3, 3, 3, 3, 3, 3, 3, 3
Let an be the number of ways to climb n stairs if a person climbing the stairs can take one stair or two stairs at a time. Identify the number of ways the person who can take one stair or two stairs at a time can climb a flight of eight stairs.
34
Consider the graph K4. Identify the dimension of the adjacency matrix of the given graph.
4 X 4
Identify the degree sequence of each of the following graphs. W4
4, 3, 3, 3, 3
Find the number of primes less than 180 using the principle of inclusion-exclusion.
41 N(P′1P′2P′3P′4P′5P′6) = 178 - 89 - 59 - 35 - 25 - 16 - 13 + 29 + 17 + 12 + 8 + 6 + 11 + 8 + 5 + 4 + 5 + 3 + 2 + 2 + 1 + 1 - 5 - 4 - 2 - 2 - 2 - 1 - 1 - 1 - 0 - 1 - 1 - 0 - 0 ... + 0 N(P′1P′2P′3P′4P′5P′6) = 35 These 35 integers are all prime, as are the 6 integers in our set. Thus, there are exactly 35 + 6 = 41 prime numbers less than 180.
Consider the graph C4. Identify the dimension of the adjacency matrix of the given graph.
4X4
Find the number of ways to distribute six different toys to three different children such that each child gets at least one toy. The number of ways to distribute six different toys to three different children such that each child gets at least one toy is
540
Consider the graph K1,4. Identify the dimension of the adjacency matrix of the given graph.
5X5
Consider the graph K2,3. Identify the dimension of the adjacency matrix of the given graph. Enter the elements of the adjacency matrix of the given graph.
5X5
Consider the graph W4. Identify the dimension of the adjacency matrix of the given graph. Enter the elements of the adjacency matrix of the given graph
5X5
Consider the undirected graph. How many edges are there in the given graph?
6
Consider the undirected graph. Identify the number of vertices in the given graph.
6
Find the number of paths of length n between two different vertices in K4if n = 3. There are 7 paths of length 3 between two vertices.
7
Consider a graph having degree sequence 4, 3, 3, 2, 2. How many edges does a graph have if its degree sequence is 4, 3, 3, 2, 2? (You must provide an answer before moving to the next part.)
7 It has (4 + 3 + 3 + 2 + 2)/2 = 7 edges.
Consider the directed multigraph How many edges are there for the given graph? What is the number of in-degrees for the vertices
8 The number of edges is eight. d = 1 degree a = 2 degree c= 2 degree b = 3 degree
Consider the graph Q3. Identify the dimension of the adjacency matrix of the given graph. Enter the elements of the adjacency matrix of the given graph
8X8
Identify the following graphs: K4,4
A complete bipartite graph Km,n is a graph that has its vertex set partitioned into two subsets of m and n vertices, respectively, with an edge between two vertices if and only if one vertex is in the first subset and the other vertex is in the second subset. The complete bipartite graph K4,4has four vertices in each partite set.
Identify the following graphs: C7
A cycle Cn, n ≥ 3, consists of n vertices v1, v2, . . . , vn and edges {v1, v2}, {v2, v3}, . . . , {vn − 1, vn}, and {vn, v1}. The graph of C7 is
Suppose that there are five young women and six young men on an island. Each woman is willing to marry some of the men on the island and each man is willing to marry any woman who is willing to marry him. Suppose that Anna is willing to marry Jason, Larry, and Matt; Barbara is willing to marry Kevin and Larry; Carol is willing to marry Jason, Nick, and Oscar; Diane is willing to marry Jason, Larry, Nick, and Oscar; and Elizabeth is willing to marry Jason and Matt. Identify a matching of the young women and the young men on the island such that each young woman is matched with a young man whom she is willing to marry.
AL, BK, CJ, DN, and EM
Consider simple graphs with their adjacency matrices. Click and drag the steps to determine whether the given graphs, described by their adjacency matrices, are isomorphic.
Both the graphs are paths with three vertices. Hence, they are clearly isomorphic.
Order the steps to create an argument that the graph is not bipartite.
Consider the vertices b, c, and f. The selected vertices form a triangle. By the pigeonhole principle, at least two of the vertices must be in the same partite set of the proposed bipartition. There would be an edge joining two vertices in the same partite set, hence this graph is not bipartite.
Consider the following pair of graphs: The graphs are not isomorphic.
FALSE
Consider the given bipartite graphs. Is Wn bipartite for n ≥ 3?
FALSE
Consider the graphs, Kn, Cn, Wn, Km,n, and Qn. How many vertices and how many edges does Kn have?
It has n vertices and n(n − 1)/2 edges.
Consider the graph. Identify the true statement about the graph shown. Consider the graph. Does the graph have directed edges?
It has no multiple edges and no loops. Loops are edges that connect a vertex to itself. Two or more edges that join the same pair of vertices are called multiple edges. Does the graph have directed edges? No A directed graph (or digraph) consists of a nonempty set of vertices and a set of directed edges (or arcs). Each directed edge is associated with an ordered pair of vertices. The directed edge associated with the ordered pair (a, b) is said to start at a and end at b.
Suppose that there are five young women and six young men on an island. Each woman is willing to marry some of the men on the island and each man is willing to marry any woman who is willing to marry him. Suppose that Anna is willing to marry Jason, Larry, and Matt; Barbara is willing to marry Kevin and Larry; Carol is willing to marry Jason, Nick, and Oscar; Diane is willing to marry Jason, Larry, Nick, and Oscar; and Elizabeth is willing to marry Jason and Matt. Identify the correct statements with respect to the matching between men and women formed in the previous question.
It is a complete matching from the set of women to the set of men. It is a maximum matching, since complete matching is always a maximum matching.
Arrange the steps in correct order to show that the sum, over the set of people at a party, of the number of people a person has shaken hands with, is even. Assume that no one shakes his or her own hand.
Let the vertices of a graph be people at a party, with an edge between two vertices if two people shake hands. The degree of each vertex is the number of people with whom a person shakes hands. Using the handshaking theorem, the sum of the degrees is 2e. Hence, the sum of the number of people a person has shaken hands with is even.
What kind of graph from the following table can be used to model a highway system between major cities, where there is an edge between the vertices representing cities for each interstate highway between them?
Multigraph
What kind of graph from the following table can be used to model a highway system between major cities, where there is an edge between the vertices representing cities if there is an interstate highway between them?
Simple graph
Consider the graph K1,4. Enter the elements of the adjacency matrix of the given graph.
Since K1,4 is a complete bipartite graph with 1 + 4 = 5 vertices, one vertex is adjacent to the other four vertices. Here, vertex 1 is adjacent with vertex 2, vertex 3, vertex 4, and vertex 5. So, the elementsa1,2,a1,3,a1,4,a1,5,a2,1,a3,1,a4,1, anda5,1are 1s, and the remaining elements are 0s.
Consider a graph having degree sequence 4, 3, 3, 2, 2. The graph for which the degree sequence is 4, 3, 3, 2, 2 is
TRUE
The web can be modeled as a directed graph where each web page is represented by a vertex and where an edge starts at the web page a and ends at the web page b if there is a link on a pointing to b. This model is called the web graph. The in-degree of a vertex is the number of other web pages that have a link to it.
TRUE
The web can be modeled as a directed graph where each web page is represented by a vertex and where an edge starts at the web page a and ends at the web page b if there is a link on a pointing to b. This model is called the web graph. The out-degree of a vertex is the number of links on the web page.
TRUE
Consider the following pair of graphs: The graphs are isomorphic.
TRUE The given graphs have the same number of vertices and exhibit a one-to-one correspondence; thus, they are isomorphic to each other.
Identify the adjacency list to represent the given graph.
The adjacency list of an undirected graph is a list of the vertices of the given graph, together with a list of the vertices adjacent to each.
Consider simple graphs with their adjacency matrices. Click and drag the steps to determine whether the given graphs, described by their adjacency matrices, are isomorphic.
The first graph has four edges, and the second graph has five edges. Hence, the graphs are not isomorphic.
Consider simple graphs with their adjacency matrices. Click and drag the steps to determine whether the given graphs, described by their adjacency matrices, are isomorphic.
The first graph has four edges, and the second graph has three edges. Hence, the graphs are not isomorphic.
Suppose that there are five young women and five young men on an island. Each man is willing to marry some of the women on the island and each woman is willing to marry any man who is willing to marry her. Suppose that Sandeep is willing to marry Tina and Vandana; Barry is willing to marry Tina, Xia, and Uma; Teja is willing to marry Tina and Zelda; Anil is willing to marry Vandana and Zelda; and Emilio is willing to marry Tina and Zelda. Identify the steps to show using Hall's theorem that there is no matching of the young men and young women on the island such that each young man is matched with a young woman he is willing to marry.
The partite sets are the set of men and the set of women. We have edges from Anil to {Vandana, Zelda}, Barry to {Tina, Uma, Xia}, Emilio to {Tina, Zelda}, Sandeep to {Vandana, Tina}, Teja to {Zelda, Tina}. We do not put an edge between a man and a woman he is not willing to marry. The condition in Hall's theorem is violated by {Uma, Xia}, because these two vertices are adjacent only to Barry. Only Barry is willing to marry Uma and Xia; so, there can be no matching.
Suppose that there are five young women and six young men on an island. Each woman is willing to marry some of the men on the island and each man is willing to marry any woman who is willing to marry him. Suppose that Anna is willing to marry Jason, Larry, and Matt; Barbara is willing to marry Kevin and Larry; Carol is willing to marry Jason, Nick, and Oscar; Diane is willing to marry Jason, Larry, Nick, and Oscar; and Elizabeth is willing to marry Jason and Matt. Identify the bipartite graph used to model the possible marriages on the island. (Use the first letters of their respective names to represent the vertices in the bipartite graph.)
The partite sets are the set of women {Anna, Barbara, Carol, Diane, Elizabeth} represented by {A, B, C, D, E} and the set of men {Jason, Kevin, Larry, Matt, Nick, Oscar} represented by {J, K, L, M, N, O}. The given information in the question tells us that there are edges AJ, AL, AM, BK, BL, CJ, CN, CO, DJ, DL, DN, DO, EJ, and EM in the required model. We do not put an edge between a woman and a man she is not willing to marry.
Let G be a graph with v vertices and e edges. Let M be the maximum degree of the vertices of G, and let m be the minimum degree of the vertices of G. Identify the correct statements involved in the proof of the statement 2e/v ≥ m.
We know that 2e is the sum of the degrees of the vertices. The degree of each vertex is greater than or equal to m, and there are v vertices. The sum of the degrees of the vertices must be ≥ vm. Hence, 2e ≥ vm.Therefore, 2e/v ≥ m.
Let an be the number of bit strings of length n that do not contain three consecutive 0s. Identify the initial conditions for the recurrence relation in the previous question.
a0 = 1, a1 = 2, and a2 = 4
Solve these recurrence relations together with the initial conditions given. Identify the solution of the recurrence relation an = 6an − 1 - 8an − 2 for n ≥ 2 together with the initial conditions a0 = 4 and a1 = 10.
an = 3 · 2n + 4n
Solve these recurrence relations together with the initial conditions given. Identify the solution of the recurrence relation an = 2an − 1 - an − 2 for n ≥ 2 together with the initial conditions a0 = 4 and a1 = 1.
an = 4 - 3n
Let an be the number of bit strings of length n that do not contain three consecutive 0s. Identify a recurrence relation for an.
an = an - 1 + an - 2 + an - 3 for n ≥ 3
Consider the following pair of graphs: If the given graphs are isomorphic, then identify the correct mapping (an isomorphism) between the sets of vertices of the two graphs.
f(u1) = v1, f(u2) = v2, f(u3) = v4, f(u4) = v5, and f(u5) = v3
Consider the given bipartite graphs. For what values of n is Cn bipartite?
for all even n ≥ 4
Consider the given bipartite graphs. For what values of n is Qn bipartite?
for all n ≥ 1
For which values of n are these graphs regular? Kn
for all n ≥ 1
For which values of n are these graphs regular? Wn
for n = 3
Identify the properties of the given recurrence relations. an = an - 2
linear and homogeneous with constant coefficients and degree 2
Identify the properties of the given recurrence relations. an = 3an - 1 + 4an - 2 + 5an - 3
linear and homogeneous with constant coefficients and degree 3
Identify the properties of the given recurrence relations. an = an - 1 + an - 4
linear and homogeneous with constant coefficients and degree 4
Identify the properties of the given recurrence relations. an = 2nan - 1 + an - 2
linear and homogeneous with nonconstant coefficients
What is the general form of the particular solution guaranteed to exist by Theorem 6 of the linear nonhomogeneous recurrence relation an = 6an - 1 - 12an - 2 + 8an - 3 + F(n) if F(n) = n2n?
n^3 (p1n + p0)2^n
Identify the properties of the given recurrence relations. an = a2n - 1 + an - 2
nonlinear and homogeneous with constant coefficients
Determine whether each of these graphs is strongly connected; if not, check whether it is weakly connected
weakly connected
Arrange the steps in the correct order to find whether a simple graph with 15 vertices, each of degree 5, exists.
Let V1 be the set of even degree vertices and V2 be the set of odd degree vertices in an undirected graph with m edges. 2m=∑v∈Vdegv=∑v∈V1degv+∑v∈V2degv⇒2m=∑v∈V1degv+∑v∈V2degv2m=∑v∈Vdegv=∑v∈V1degv+∑v∈V2degv⇒2m=∑v∈V1degv+∑v∈V2degv Clearly ∑v∈V1degv∑v∈V1degv is an even number. The left-hand side of the previous equation is even and one of the terms in the sum on the right-hand side is even, so their difference 2m−∑v∈V1degv=∑v∈V2degv2m−∑v∈V1degv=∑v∈V2degv must be even. Since the sum is even and all the terms are odd, there must be an even number of such terms. Hence, in any graph the number of odd degree vertices are even. Therefore, there exists no graph with 15 vertices each of degree 5.
Enter the elements of the adjacency matrix of the given graph. Consider the graph K4.
Since K4 is a complete graph with four vertices, all the vertices are adjacent to each other. Vertex 1 is adjacent to vertex 2, vertex 3, and vertex 4. Each vertex is not adjacent to itself. So, the elements a1,1, a2,2, a3,3, and a4,4 are 0s, and the remaining elements are 1s.
Consider the undirected graph. Identify the isolated vertex from the given graph.
d
What is the general form of the particular solution guaranteed to exist by Theorem 6 of the linear nonhomogeneous recurrence relation an = 6an - 1 - 12an - 2 + 8an - 3 + F(n) if F(n) = 3?
p0
What is the general form of the particular solution guaranteed to exist by Theorem 6 of the linear nonhomogeneous recurrence relation an = 6an - 1 - 12an - 2 + 8an - 3 + F(n) if F(n) = (-2)n?
p0(-2)^n
What is the general form of the particular solution guaranteed to exist by Theorem 6 of the linear nonhomogeneous recurrence relation an = 6an - 1 - 12an - 2 + 8an - 3 + F(n) if F(n) = n^2?
p2n^2 + p1n + p0
Find the strongly connected components of each of these graphs.
{a, b, e}, {c}, and {d}
Find the strongly connected components of each of these graphs.
{a}, {b}, {c, d, e}, and {f}
Find the number of elements in A1 ∪ A2 ∪ A3 if there are 100 elements in A1, 1000 in A2, and 10,000 in A3 if A1 ⊆ A2 and A2 ⊆ A3.
10000 It is given that A1 ⊆ A2 and A2 ⊆ A3, which implies that A1 ⊆ A3; we can conclude that A1 ∩ A2= A1, A2 ∩ A3 = A2, and A1 ∩ A3 = A1. We know that |A1 ∪ A2 ∪ A3| = |A1| + |A2| + |A3| - |A1 ∩ A2| - |A2 ∩ A3| - |A1 ∩ A3| + |A1 ∩ A2 ∩ A3|. |A1 ∪ A2 ∪ A3| = |A1| + |A2| + |A3| - |A1| - |A2| - |A1| + |A1| = |A3| = 10,000
How many nonisomorphic simple graphs are there with n vertices for the following values of n? When n = 4, there are (answer) nonisomorphic simple graphs.
11 When n = 4, there are 11 nonisomorphic simple graphs.
Consider the directed graph shown in the figure. Determine the sum of the in-degrees of the vertices for the given graph
13
How many elements are in A1 ∪ A2 if there are 12 elements in A1, 18 elements in A2, and |A1 ∩ A2| = 6? The number of elements in A1 ∪ A2 is .
24 Using the principle of inclusion-exclusion, we have |A1 ∪ A2| = |A1| + |A2| - |A1 ∩ A2|. Therefore, |A1 ∪ A2| = 12 + 18 - 6 = 24.
Identify the degree sequence of each of the following graphs. K2, 3
3, 3, 2, 2, 2
Find the number of positive integers not exceeding 10,020 that are not divisible by 3, 4, 7, or 11.
3904
How many nonisomorphic simple graphs are there with n vertices for the following values of n? When n = 3, there are (answer) nonisomorphic simple graphs.
4 When n = 3, there are 4 nonisomorphic simple graphs.
How many permutations of the 26 letters of the English alphabet do not contain any of the strings fish, rat, or bird? (answer) × 1026 (Enter the answer accurate to two decimal places.)
4.03 26! - (23! + 24! + 23! - 21!) = 4.0262 × 10^26.
Find the number of paths of length n between two different vertices in K4if n = 5. There are 61 paths of length 5 between two vertices.
61
Let an be the number of bit strings of length n that do not contain three consecutive 0s. Identify the number of bit strings of length seven that do not contain three consecutive 0s.
81
The word rock can refer to a type of music or to something from a mountain. Identify the word graph for these nouns: rock, boulder, jazz, limestone, gravel, folk, bachata, pumice, granite, tango, klezmer, slate, shale, classical, pebbles, sand, rap, and marble. Two vertices are connected by an undirected edge if the nouns they represent have similar meaning.
An edge is drawn when the words have related meanings. Thus, we obtain a word graph as follows:
Identify the following graphs: Q4
An n-dimensional hypercube, or n-cube, denoted by Qn, is a graph that has vertices representing the 2n bit strings of length n. Two vertices are adjacent if and only if the bit strings that they represent differ in exactly one bit position. We obtain the graph of Q4 by taking two copies of Q3 and by joining the corresponding vertices. Q3 is constructed from Q2 by drawing two copies of Q2 as the top and bottom faces of Q3. The graph of Q4 is
A model for the number Ln of lobsters caught per year is based on the assumption that the number of lobsters caught in a year is the average of the number caught in the two previous years. Identify a recurrence relation for Ln.
Ln = (1/2)Ln − 1 + (1/2)Ln − 2
What kind of graph from the following table can be used to model a highway system between major cities, where there is an edge between the vertices representing cities for each interstate highway between them and there is a loop at the vertex representing a city if there is an interstate highway that circles this city?
Pseudograph
Consider a graph model that represents traditional marriages between men and women. The set of vertices represents a set of people, and two vertices are joined by an edge if two people were ever married. Identify the special property of the graph model.
There are two types of vertices, i.e., men and women, and every edge joins the vertices of opposite types.
Consider the given graphs: Click and drag the statements in correct order either to show that these graphs are not isomorphic or to find an isomorphism between them using paths.
There are two vertices in each graph that are not in any cycles of size 4. So try to construct an isomorphism that matches them, say u1 ↔ v2 and u8 ↔ v6. Now u1 is adjacent to u2 and u3, and v2 is adjacent to v1 and v3, so u2 ↔ v1 nd u3 ↔ v3. Since u4 is the other vertex adjacent to u3 and v4 is the other vertex adjacent to v3, u4 ↔ v4. Proceeding along similar lines, complete the bijection with u5 ↔ v8, u6 ↔ v7, and u7 ↔ v5. Having thus been led to the only possible isomorphism, check that the 12 edges of G correspond to the 12 edges of H; hence the two graphs are isomorphic.
Let G be a graph with v vertices and e edges. Let M be the maximum degree of the vertices of G, and let m be the minimum degree of the vertices of G. Show that 2e/v ≤ M.
We know that 2e is the sum of the degrees of the vertices, the degree of each vertex is ≤ M, and there are v vertices. The sum of the degrees of the vertices must be ≤ vM. Hence 2e ≤ vM.
Identify the following graphs: W7
We obtain a wheel Wn when we add an additional vertex to a cycle Cn, for n ≥ 3, and connect this new vertex to each of the n vertices in Cn, by new edges. The graph of W7 is
A country uses coins with values of 1 peso, 2 pesos, 5 pesos, and 10 pesos and bills with values of 5 pesos, 10 pesos, 20 pesos, 50 pesos, and 100 pesos as its currency. Find a recurrence relation for the number of ways to pay a bill of n pesos if the order in which the coins and bills are paid matters.
an = an - 1 + an - 2 + 2an - 5 + 2an - 10 + an - 20 + an - 50 + an - 100, for n ≥ 100
Identify the solution to an = 2an − 1 + an − 2 − 2an − 3 with a0 = 3, a1 = 6, and a2 = 0 for all integers n ≥≥ 3.
an = 6 - 2( -1)n - 2n
A bus driver pays all tolls using only nickels and dimes by throwing one coin at a time into the mechanical toll collector. Find a recurrence relation for the number of different ways the bus driver can pay a toll of n cents, where the order in which the coins are used matters.
an = an - 1 + an - 2, where an is the number of ways to pay a toll of 5n cents, n ≥ 2.
Let an be the number of ways to climb n stairs if a person climbing the stairs can take one stair or two stairs at a time. Identify a recurrence relation for an.
an = an −1 + an − 2 for n ≥ 2
Consider the following pair of graphs:
f(u1) = v1, f(u2) = v3, f(u3) = v2, f(u4) = v5, and f(u5) = v4
For which values of n are these graphs regular? Qn
for all n ≥ 1
For which values of n are these graphs regular? Cn
for all n ≥ 3
Consider the given bipartite graphs. For what value of n is Kn bipartite?
for n = 2
Identify the properties of the given recurrence relations. an = an - 1 + 2
linear and nonhomogeneous with constant coefficients
Identify the properties of the given recurrence relations. an = an - 1 + n
linear and nonhomogeneous with constant coefficients
What is the general form of the particular solution guaranteed to exist by Theorem 6 of the linear nonhomogeneous recurrence relation an = 6an - 1 - 12an - 2 + 8an - 3 + F(n) if F(n) = 2n?
n^3 p02^n
Determine whether each of these graphs is strongly connected; if not, check whether it is weakly connected
neither strongly nor weakly connected
Identify the solution of the recurrence relation an = 6an - 1 - 12an - 2 + 8an - 3 with a0 = -5, a1 = 4, and a2 = 88.
-5 · 2n + n · 2n - 1 + 13n2 · 2n - 1
Find the number of elements in A1 ∪ A2 ∪ A3 if there are 120 elements in each set and if there are 40 common elements in each pair of sets and 24 common elements in all three sets.
264 We use the principle of inclusion-exclusion for three sets, i.e., |A1 ∪ A2 ∪ A3| = |A1| + |A2| + |A3| - |A1 ∩ A2| - |A1 ∩ A3| - |A2 ∩ A3| + |A1 ∩ A2 ∩ A3|. If the sets are pairwise disjoint, then |A1 ∩ A2| = |A1 ∩ A3| = |A2 ∩ A3| = 40 and |A1 ∩ A2 ∩ A3| = 24. |A1 ∪ A2 ∪ A3| = 120 + 120 + 120 - 40 - 40 - 40 + 24 = 264
A survey of households in the United States reveals that 94% have at least one television set, 95% have telephone service, and 93% have telephone service and at least one television set. What percentage of households in the United States have neither telephone service nor a television set?
4 We may treat the percentages as if they were cardinalities, as if the population were exactly 100. Let V be the set of households with television sets, and let P be the set of households with telephones; thus, the cardinalities are |V| = 94, |P| = 95, and |V ∩ P| = 93. Using the principle of inclusion-exclusion, we have |V ∪ P| = |V| + |P| - |V ∩ P| = 94 + 95 - 93 = 96. Thus, there are 96 households that have both television sets and telephones. We need the number of households that have neither televisions nor telephones, which is obtained by subtracting |V ∪ P| from 100. 100 - |V ∪ P| = 4
Find the number of elements in A1 ∪ A2 ∪ A3 if there are 100 elements in A1, 1000 in A2, and 10,000 in A3 if there are two elements common to each pair of sets and one element in all three sets.
11095 It is given that |A1 ∩ A2| = |A2 ∩ A3| = |A1 ∩ A3| = 2 and |A1 ∩ A2 ∩ A3| = 1. We know that |A1 ∪ A2 ∪ A3| = |A1| + |A2| + |A3| - |A1 ∩ A2| - |A2 ∩ A3| - |A1 ∩ A3| + |A1 ∩ A2 ∩ A3|. |A1 ∪ A2 ∪ A3| = 100 + 1000 + 10,000 - 2 - 2 - 2 + 1 = 11,095
n a survey of 270 college students, it is found that 64 like brussels sprouts, 94 like broccoli, 58 like cauliflower, 26 like both brussels sprouts and broccoli, 28 like both brussels sprouts and cauliflower, 22 like both broccoli and cauliflower, and 14 like all three vegetables. How many of the 270 students do not like any of these vegetables?
116 Let A, B, and C be the sets of college students who like brussels sprouts, broccoli, and cauliflower, respectively. From the given data, we have |A| = 64, |B| = 94, |C| = 58, |A ∩ B| = 26, |A∩ C| = 28, |B ∩ C| = 22, and |A ∩ B ∩ C| = 14. We know that |A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|. |A ∪ B ∪ C| = 64 + 94 + 58 - 26 - 28 - 22 + 14 = 154 The number of students who do not like any of these vegetables can be determined by subtracting |A ∪ B ∪ C| from the total number of college students taken for the survey, i.e., 270. Thus, we have 270 - 154 = 116.
Determine the sum of the out-degrees of the vertices for the given graph.
13 The out-degree of v is the number of edges with v as its initial vertex. The out-degree of the vertices a, b, c, d, and e are 1, 5, 5, 2, and 0, respectively. The sum of the out-degrees is 1 + 5 + 5 + 2 + 0 = 13.
How many elements are in A1 ∪ A2 if there are 12 elements in A1, 18 elements in A2, and A1 ⊆ A2? The number of elements in A1 ∪ A2 is .
18 It is given that A1 ⊆ A2; thus, A1 ∩ A2 = A1. Using the principle of inclusion-exclusion, we have |A1 ∪ A2| = |A1| + |A2| - |A1 ∩ A2|. Therefore, |A1 ∪ A2| = 12 + 18 - 12 = 18.
How many terms are there in the formula for the number of elements in the union of 11 sets given by the principle of inclusion-exclusion?
2047 The principle of inclusion-exclusion gives a formula for the number of elements in the union of n sets for every positive integer n. In the formula for n sets, there are C(n, 1) + C(n, 2) + C(n, 3) + ... + C(n, n) = 2n - C(n, 0) terms on the right-hand side. For n = 11, there are 2^11 - C(11, 0) = 2^11 - 1 = 2047 terms on the right-hand side.
How many elements are in the union of four sets if each of the sets has 106 elements, each pair of the sets shares 50 elements, each three of the sets share 25 elements, and there are 5 elements in all four sets?
219 Let A, B, C, and D be four sets. From the given data, we have |A| = |B| = |C| = |D| = 106, |A ∩ B| = |A∩ C| = |A ∩ D| = |B ∩ C| = |B ∩ D| = |C ∩ D| = 50, |A ∩ B ∩ C| = |A ∩ C ∩ D| = |B ∩ C ∩ D| = |A ∩ B ∩ D| = 25, and |A ∩ B ∩ C ∩ D| = 5. We know that |A ∪ B ∪ C ∪ D| = |A| + |B| + |C| + |D| - |A ∩ B| - |A ∩ C| - |A ∩ D| - |B ∩ C| - |B ∩ D| - |C ∩ D| + |A ∩ B ∩ C| + |A ∩ C ∩ D| + |B ∩ C ∩ D| + |A ∩ B ∩ D| - |A ∩ B ∩ C ∩ D|. |A ∪ B ∪ C ∪ D| = 4 × 106 - 6 × 50 + 4 × 25 - 5 = 219
Find the number of elements in A1 ∪ A2 ∪ A3 if there are 120 elements in each set and if there are 40 common elements in each pair of sets and no common elements in all three sets.
240 We use the principle of inclusion-exclusion for three sets, i.e., |A1 ∪ A2 ∪ A3| = |A1| + |A2| + |A3| - |A1 ∩ A2| - |A1 ∩ A3| - |A2 ∩ A3| + |A1 ∩ A2 ∩ A3|. |A1 ∩ A2| = |A1 ∩ A3| = |A2 ∩ A3| = 40 and |A1 ∩ A2 ∩ A3| = 0. Thus, |A1 ∪ A2 ∪ A3| = |A1| + |A2| + |A3| - |A1 ∩ A2| - |A1 ∩ A3| - |A2 ∩ A3|. |A1 ∪ A2 ∪ A3| = 120 + 120 + 120 - 40 - 40 - 40 = 240
In how many ways can a 2 × n rectangular checkerboard be tiled using 1 × 2 and 2 × 2 pieces?
tn=2n+13+(−1)n3tn=2n+13+(-1)n3
How many elements are in A1 ∪ A2 if there are 12 elements in A1, 18 elements in A2, and A1 ∩ A2 = ∅? The number of elements in A1 ∪ A2 is .
30 Using the principle of inclusion-exclusion, we have |A1 ∪ A2| = |A1| + |A2| - |A1 ∩ A2|. Since A1 ∩ A2 = ∅, |A1 ∩ A2| = 0. Therefore, |A1 ∪ A2| = 12 + 18 - 0 = 30.
How many elements are in the union of four sets if the sets have 60, 60, 70, and 80 elements, respectively, each pair of the sets has 5 elements in common, each triple of the sets has 1 common element, and no element is common in all four sets?
244 Let the four sets be A, B, C, and D. By the principle of inclusion-exclusion, we have |A ∪ B ∪ C ∪ D| = |A| + |B| + |C| + |D| - |A ∩ B| - |A ∩ C| - |A ∩ D| - |B ∩ C| - |B ∩ D| - |C ∩ D| + |A ∩ B ∩ C| + |A ∩ C ∩ D| + |B ∩ C ∩ D| + |A ∩ B ∩ D| - |A ∩ B ∩ C ∩ D|. |A ∪ B ∪ C ∪ D| = 60 + 60 + 70 + 80 - (6 × 5) + (4 × 1) - 0 = 244
How many bit strings of length eight do not contain six consecutive 0s?
248 Let A be the set of outcomes in which the first six digits of the bit strings are zeros, let B be the set of outcomes in which the middle six digits of the bit strings are zeros, and let C be the set of outcomes in which the last six digits of the bit strings are zeros. As six digits in each bit string of length eight in the sets A, B, and C have six consecutive zeros, each of the remaining two digits can have the value 0 or 1; these two digits can be chosen in 2 · 2 = 4 ways. Thus, we have |A| = |B| = |C| = 4. The set A ∩ B consists of those bit strings in which the first six and middle six digits are zeros; so, the first seven digits are zeros. Thus, we have to choose only the last digit of the string. This can be done in two ways. Therefore, |A ∩ B| = 2. Similarly, |B ∩ C| = 2. The set A ∩ C consists of those bit strings in which the first and the last six digits are zeros; so, all the digits in the bit string are zeros, i.e., |A ∩ C| = 1. Similarly, the set A ∩ B ∩ C consists of those bit strings in which the first, middle, and last six digits are zeros. Using the same logic, we get |A ∩ B ∩ C| = 1. We know that |A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|. |A ∪ B ∪ C| = 4 + 4 + 4 - 2 - 1 - 2 + 1 = 8 There are 28 bit strings of length eight. So, there are 28 - 8 = 256 - 8 = 248 bits strings that do not contain six consecutive 0s.
A marketing report concerning personal computers states that 650,000 owners will buy a printer for their machines next year and 1,250,000 will buy at least one software package. If the report states that 1,550,000 owners will buy either a printer or at least one software package, how many will buy both a printer and at least one software package?
350000 Let P be the set of owners who will buy a printer, |P| = 650,000. Let S be the set of owners who will buy a software package, |S| = 1,250,000. P ∪ S represents the set of owners who will buy either a printer or at least one software package, |P ∪ S| = 1,550,000. Using the principle of inclusion-exclusion, we have |P ∪ S| = |P| + |S| - |P ∩ S|. As we require |P ∩ S|, we can rewrite the equation and solve as follows: |P ∩ S| = |P| + |S| - |P ∪ S| = 650,000 + 1,250,000 - 1,550,000 = 350,000
Find the number of elements in A1 ∪ A2 ∪ A3 if there are 120 elements in each set and if the sets are pairwise disjoint.
360 We use the principle of inclusion-exclusion for three sets, i.e., |A1 ∪ A2 ∪ A3| = |A1| + |A2| + |A3| - |A1 ∩ A2| - |A1 ∩ A3| - |A2 ∩ A3| + |A1 ∩ A2 ∩ A3|. If the sets are pairwise disjoint, then |A1 ∩ A2| = |A1 ∩ A3| = |A2 ∩ A3| = |A1 ∩ A2 ∩ A3| = 0. Thus, |A1 ∪ A2 ∪ A3| = |A1| + |A2| + |A3|. |A1 ∪ A2 ∪ A3| = 120 + 120 + 120 = 360
There are 345 students at a college who have taken a course in calculus, 212 who have taken a course in discrete mathematics, and 188 who have taken courses in both calculus and discrete mathematics. How many students have taken a course in either calculus or discrete mathematics?
369 Let C be the set of students who have taken a course in calculus; thus, |C| = 345. Let D be the set of students who have taken a course in discrete mathematics; thus, |D| = 212. Then, C ∩ Drepresents the set of students who have taken courses in both calculus and discrete mathematics; thus, |C ∩ D| = 188. Using the principle of inclusion-exclusion, we have |C ∪ D| = |C| + |D| - |C ∩ D|. By substituting the values of |C|, |D|, and |C ∩ D|, we get |C ∪ D| = 345 + 212 - 188 = 369
There are 345 students at a college who have taken a course in calculus, 212 who have taken a course in discrete mathematics, and 188 who have taken courses in both calculus and discrete mathematics. Find the number of students who have taken a course in either calculus or discrete mathematics. The number of students who have taken a course in either calculus or discrete mathematics is .
369 Let C be the set of total students and D be the set of the students who have taken a course in discrete mathematics; then, ∣∣C∪D∣∣=∣∣C∣∣+∣∣D∣∣−∣∣C∩D∣∣=345+212−188=369C∪D=C+D-C∩D=345+212-188=369 .
Find the number of positive integers not exceeding 1000 that are either the square or the cube of an integer.
38 Let A be the set of integers that are squares not exceeding 1000 and B be the set of integers that are cubes not exceeding 1000. Thus, A ∩ B represents the set of integers that are both squares and cubes not exceeding 1000 and A ∪ B represents the set of integers that are either squares or cubes not exceeding 1000. There are ⌊1000‾‾‾‾‾√⌋=311000=31 squares and ⌊1000‾‾‾‾‾√3⌋=1010003=10cubes. Furthermore, there are ⌊1000‾‾‾‾‾√6⌋=310006=3 integers that are both squares and cubes (i.e., sixth powers). We know that |A ∪ B| = |A| + |B| - |A ∩ B|. |A ∪ B| = 31 + 10 - 3 = 38
There are 2510 computer science students at a school. Of these, 1876 have taken a course in Java, 999 have taken a course in Linux, and 345 have taken a course in C. Further, 876 have taken courses in both Java and Linux, 231 have taken courses in both Linux and C, and 290 have taken courses in both Java and C. If 189 of these students have taken courses in Linux, Java, and C, how many of these 2510 students have not taken a course in any of these three programming languages?
498 Let J, L, and C represent the sets of students who have taken courses in Java, Linux, and C, respectively. It is given that the total number of students in the school is 2510. From the given data, we have |J| = 1876, |L| = 999, |C| = 345, |J ∩ L| = 876, |L ∩ C| = 231, |J ∩ C| = 290, and |J ∩ L ∩ C| = 189. We know that |J ∪ L ∪ C| = |J| + |L| + |C| - |J ∩ L| - |L ∩ C| - |J ∩ C| + |J ∩ L ∩ C|. |J ∪ L ∪ C| = 1876 + 999 + 345 - 876 - 231 - 290 + 189 = 2012 As there are 2510 students altogether, we have 2510 - 2012 = 498 students who have not taken any courses.
How many permutations of the 10 digits either begin with the three digits 987, contain the digits 45 in the fifth and sixth positions, or end with the three digits 123?
50138 Let A be the set of 10 digits that begin with the three digits 987, B be the set of 10 digits that contain 45 in the fifth and sixth positions, and C be the set of 10 that digits end with the three digits 123. If the 10 digits begin with 987, then there are (10 - 3)! = 7! permutations. Similarly, there are (10 - 2)! = 8! permutations that have 45 in the fifth and sixth positions and there are (10 - 3)! = 7! permutations that end with digits 123. Further, there are (10 - 5)! = 5! permutations that begin with 987 and have 45 in the fifth and sixth positions; (10 - 6)! = 4! permutations that begin with 987 and end with 123; and (10 - 5)! = 5! permutations that have 45 in the fifth and sixth positions, and end with 123. Using the principle of inclusion-exclusion on the three sets, we have |A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|. |A ∪ B ∪ C| = 7! + 8! + 7! - 5! - 4! - 5! + 2! = 50,138 permutations
A bus driver pays all tolls using only nickels and dimes by throwing one coin at a time into the mechanical toll collector. In how many different ways can a driver pay a toll of 45 cents?
55 A driver pays all tolls using only nickels and dimes. As an is the number of ways to pay a toll of 5n cents, we must find a9, which gives us the number of ways to pay a toll of 5 × 9 = 45 cents. We can obtain the solution using the recurrence relation an = an - 1 + an - 2 with a0 = a1 = 1 as follows: a2 = a1 + a0 = 1 + 1 = 2 a3 = a2 + a1 = 2 + 1 = 3 a4 = a3 + a2 = 3 + 2 = 5 a5 = a4 + a3 = 5 + 3 = 8 a6 = a5 + a4 = 8 + 5 = 13 a7 = a6 + a5 = 13 + 8 = 21 a8 = a7 + a6 = 21 + 13 = 34 a9 = a8 + a7 = 34 + 21 = 55
Find the number of positive integers not exceeding 1000 that are not divisible by 3, 17, or 35.
610 We first compute the number of positive integers not exceeding 1000 that are divisible by at least one of 3, 17, and 35. We use the inclusion-exclusion principle on the three sets. Let A be the set of positive integers not exceeding 1000 that are divisible by 3, B be the set of positive integers not exceeding 1000 that are divisible by 17, and C be the set of positive integers not exceeding 1000 that are divisible by 35. We determine |A ∪ B ∪ C| as follows: |A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C| ∣∣A∪B∪C∣∣=⌊10003⌋+⌊100017⌋+⌊100035⌋−⌊10003·17⌋−⌊10003·35⌋−⌊100017·35⌋+⌊10003·17·35⌋|A∪B∪C|=10003+100017+100035-10003·17-10003·35-100017·35+10003·17·35 |A ∪ B ∪ C| = 333 + 58 + 28 - 19 - 9 - 1 + 0 = 390 The number of positive integers not exceeding 1000 that are divisible by at least one of 3, 17, and 35 is 390; so, the number of positive integers not exceeding 1000 that are not divisible by 3, 17, or 35 numbers is 1000 - 390 = 610.
Find the number of positive integers not exceeding 104 that are not divisible by 5 or by 7.
72 To find the number of positive integers not exceeding 104 that are not divisible by 5 or by 7, we will subtract from 104 the number of positive integers that are divisible. Of the positive integers not exceeding 104, there are ⌊1045⌋1045 integers that are divisible by 5 and ⌊1047⌋1047 that are divisible by 7. As 5 and 7 are relatively prime, the integers that are divisible by both 5 and 7 are those that are divisible by 5 · 7. Consequently, there are ⌊1045·7⌋1045·7 positive integers not exceeding 104 that are divisible by both 5 and 7. It follows that there are 104−⌊104/5⌋−⌊104/7⌋+⌊104/5·7⌋104-104/5-104/7+104/5·7 = 104 - 20 - 14 + 2 = 104 - 32 = 72 positive integers that are not divisible by 5 or by 7.
Find the number of positive integers not exceeding 150 that are either odd or the square of an integer.
81 Clearly, there are 75 odd positive integers not exceeding 150 (half of the 150 numbers are odd), and there are 12 squares. Furthermore, half of these squares are odd. Thus, we compute the cardinality of the set in the question to be 75 + 12 - 6 = 81.
How many students are enrolled in a course either in calculus, discrete mathematics, data structures, or programming languages at a school if there are 514, 292, 312, and 344 students in these courses, respectively; 14 in both calculus and data structures; 213 in both calculus and programming languages; 211 in both discrete mathematics and data structures; 43 in both discrete mathematics and programming languages; and no student may take calculus and discrete mathematics, or data structures and programming languages, concurrently?
981 Let C, D, S, and P be the sets of students enrolled in courses calculus, discrete mathematics, data structures, and programming languages, respectively. From the given data, we have |C| = 514, |D| = 292, |S| = 312, |P| = 344, |C ∩ S| = 14, |C ∩ P| = 213, |D∩ S| = 211, and |D ∩ P| = 43. It is given that no student may take calculus and discrete mathematics, or data structures and programming languages, concurrently. Thus, all the intersections involving the intersections of either C and D or S and P are empty. Thus, we have |C ∩ D| = |S ∩ P| = |C ∩ D ∩ S| = |C ∩ D ∩ P| = |C ∩ S ∩ P| = |D ∩ S ∩ P| = |C ∩ D ∩ S ∩ P| = 0 |C ∪ D ∪ S ∪ P| = |C| + |D| + |S| + |P| - |C ∩ D| - |C ∩ S| - |C ∩ P| - |D ∩ S| - |D ∩ P| - |S ∩ P| + |C ∩ D∩ S| + |C ∩ D ∩ P| + |C ∩ S ∩ P| + |D ∩ S ∩ P| - |C ∩ D ∩ S ∩ P| |C ∪ D ∪ S ∪ P| = 514 + 292 + 312 + 344 - 0 - 14 - 213 - 211 - 43 - 0 + 0 + 0 + 0 + 0 - 0 = 981
Identify the call graph for a set of seven telephone numbers 555-0011, 555-1221, 555-1333, 555-8888, 555-2222, 555-0091, and 555-1200 if there were three calls from 555-0011 to 555-8888 and two calls from 555-8888 to 555-0011, two calls from 555-2222 to 555-0091, two calls from 555-1221 to each of the other numbers, and one call from 555-1333 to each of 555-0011, 555-1221, and 555-1200.
Let us consider the last four digits of the telephone numbers as vertices and arrive at the graph. The number of edges from one vertex to the another is as shown in the following table, where the numbers in the first column represent "calls from" and the numbers in the first row represent "calls to":
A model for the number Ln of lobsters caught per year is based on the assumption that the number of lobsters caught in a year is the average of the number caught in the two previous years. Identify the value of Ln if 100,000 lobsters were caught in year 1 and 300,000 were caught in year 2.
Ln = (800000/3)(−1/2)n + (700000/3)
What is the number of edges in the given graph? Find the relation between the number of edges, and the in-degree and out-degree
The number of edges is 13. The number of edges, the sum of the in-degrees, and the sum of the out-degrees are all equal. The number of edges is 13. The in-degree of the vertices a, b, c, d, and e are 6, 1, 2, 4, and 0, respectively. The sum of the in-degrees is 6 + 1 + 2 + 4 + 0 = 13. The out-degree of the vertices a, b, c, d, and e are 1, 5, 5, 2, and 0, respectively. The sum of the out-degrees is 1 + 5 + 5 + 2 + 0 = 13.
Consider the undirected graph. Which is the pendant vertex in the given graph?
c A vertex is pendant if and only if it has degree one. Since there is only one edge incident with the vertex c, it is pendant.
Consider the undirected graph Identify the degree of each vertex of the given graph.
deg(a) = 2 deg(b) = 4 deg(c) = 1 deg(d) = 0 deg(e) = 2 deg(f) = 3
What is the general form of the particular solution guaranteed to exist by Theorem 6 of the linear nonhomogeneous recurrence relation an = 6an - 1 - 12an - 2 + 8an - 3 + F(n) if F(n) = n22n?
n^3(p2n^2 + p1n + p0)2^n
Consider the graph. Identify the type of graph. Does the graph have directed edges?
simple graph NO
