5) Let G be the simple graph on the set V(G) = {p € N : p is prime, p = 1 mod 4, and p < 100}where uv E E(G) if and only if u + v and u = x has a solution modulo v. Draw this graph andanswer the following questions.(a) What is the independence number?(b) How many components does it have?(c) What is the domination number? (Size of the smallest set where every vertex is either in oradjacent to something in the set.)(d) What is the girth?(e) What is the clique number? (Size of the largest clique.)

Answered: Image /qna-images/answer/0f7ffeb0-2d06-4f44-9a57-be9d0569cca8.jpg

Answered: Let G be the simple graph on the set…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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9.
If o is a k-cycle then the order of o is k. On the otherhand, if 7 is a product of disjoint cycles o1, 02, 03 .m such that the order of o; is k; then the order of 7 is given by lem(k1, k2, kg, ..., km). Find the order each of the following. (a) (1 32 4) e S4. (b) (1 4 3)(2 5 6 7) E S7. (c) (1 43 6)(2 9)(5 7 8) E Sg. (d) (1 7)(2 6)(3 5 4) o (1 2 3)(4 5 6 7) e S,
This question is from the subject Discrete Mathematics 2, where it involves basic set theory, combinatorics, graph theory etc.
This question is from the subject Discrete Mathematics 2, where it involves basic set theory, combinatorics, graph theory etc.
For 1<= m <= 11 , let Gm be the graph with vertex set V={0,1,2,3,4,5,6,7,8,9,10,11} and where vertices u and w are adjacent if and only if w-u=m modulo 12 or u-w=m modulo 12. We observe that G1=C12, a twelve-cycle. A. For what values of m is Gm connected? B. What are the possible numbers of components of Gm?
4.1.12. Let n, k be positive integers with n even, k odd, and n > k > 1. Let G be the k- regular simple graph formed by placing n vertices on a circle and making each vertex adjacent to the opposite vertex and to the (k – 1)/2 nearest vertices in each direction. Prove that < (G) = k. (Harary (1962a])
For all n>= 1, then P(n) = n-n is divisible by 3 Select one: O True OFalse Find the value of x in the following table: 1234 Vertices In-Degree 3143 Select one: Out-Degree 4 2 X 2 O a. 3. Ob. 1. OC 4. Od. 2. for all n 1, then P(n)= (n n) is even Select one Tue Felse
Let n, k EN and define the family of graphs G (V, Ek) on the vertex set V,= (1,2,... ,n} and with edge set Enk= {(i,j) EV, x V: i j (mod k)}. In other words, two vertices i and j are adjacent if they are congruent modulok. (a) Draw G42 and G73 and show they are both disconnected. (b) Show that G is connected if and only if k= 1. (c) Show that Gis the empty graph if and only if n
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6.1.8 (a) Suppose that |A|| = 3, and |B| = 4. What are the minimum and maximum values for the cardinalities |(A × B) N (B × A)| and |(A × B) U (B × A)|? (b) More generally, suppose that |A|| = m, |B| = n and |AN B|| = c. What are the above cardinalities?

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