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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Use the formula deg(v) = 2|E(G)| to find the number of edges of the following vЄV (G) graphs. Classify (count) the vertices by number of neighbors. (a) V(G) = [100]. Edges: for all n and m in [100], n ‡ m, n is adjacent to m if and only if |nm| ≤ 4. (b) V(G) = [10] × [10]. Edges: for all (a, b) and (c,d) in [10] × [10], (a, b) ‡ (c,d), (a, b) is adjacent to (c,d) if and only if a = c or b = d. (c) V(G) = [10] × [10]. Edges: for all (a, b) and (c,d) in [10] × [10], (a, b) ‡ (c,d), (a, b) is adjacent to (c,d) if and only if |ac| + |bd| = 1. (d) V (G) = [10] × [10]. Edges: for all (a, b) and (c,d) in [10] × [10], (a, b) is adjacent to (c,d) if and only if |a - c + b-d ≤ 2. (a, b) ‡ (c,d),
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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?
(5) Use strong math induction to prove if the sum of the interior angles of a triangle is 180° then the sum of the interior angles of any convex n-gon is (n – 2) · 180°, for all n > 3. Definition: For n E N,n > 3 a convex n-gon is the n -sided figure created when successive vertices from the set of points {v1, V2, Vn} are connected by non-overlapping segments v,V2, V½V3, ·…, VnVị such that no ... interior angle exceeds 180°, where the interior angles are the angles whose rays are successive sides. Remark: Rectangles, trapezoids, hexagons and octagons are common examples of n -gons. V2 This 4-gon is not convex because the interior angle at V2 is greater than 180°.
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2.1 Let sets A, B, and C be fuzzy sets defined on real numbers by the membership functions 1 pHa(x) =+10" 1 H(x) = 10* Hi(x) x+1 Determine mathematical membership functions and graphs of each of the followings : а) AU B, BоC, b) AUBUC, AOBOC c) Anī, BUC d) AOB, ĀU B 2.2 Show the two fuzzy sets satisfy the De Morgan's Law. 1 HA(x) = 1+(x-10) 1 HB(x) = 1+x 2.3 Show that the following sets don't satisfy the law of contradiction and the law of excluded middle. a) H,(x) =- 1+x 60 / 340 b) 4 3D {(а, 0.4), (b, 0.5), (с, 0.9), (d, I)} 2.4 Determine complements, unions, and intersections of the following sets by using Yager's operators for o = 1, 2 a) A = {(a, 0.5), (b, 0.9), (c, 0.1), (d, 0.5)} 1 b) дл(х) —. 1+x 2.5 Compute the complements of the following sets by Yager's complements w = 1, 2. а) А — {(а, 0.5), (b, 0.9), (с, 0.3)} 1 b) H,(x) = 1+(x–1) 52 2. The Operation of Fuzzy Set 2.6 Compute the complements of the following sets by using Probabilistic, Bounded, Drastic, and Hamacher…

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