1.2.19. Let and s be natural numbers. Let G be the simple graph with vertex set Vo... V„−1 such that v; ↔ v; if and only if |ji| Є (r,s). Prove that S has exactly k components, where k is the greatest common divisor of {n, r,s}.
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- The graph of wheel denoted by W, is obtained when an additional vertex is added to cycle Cn, for n > 3, and connect this new vertex to each of n vertices by new edges. Match between each statement (a)- (d)) and a graph ((1)-(5)) such that the chosen graph satisfies the statement. (1) C, (2) C10 (3) W, (4) W, (5) W 10 A graph with the sum of degrees is 28. Choose... A simple and bipartite graph Choose... A simple graph with Hamiltonian circuit and vertex of degree 10. Choose... : A regular graph and not bipartite graph Choose...1.2.19. Let and s be natural numbers. Let G be the simple graph with vertex set Vo... V„−1 such that v; ↔ v; if and only if |ji| Є (r,s). Prove that S has exactly k components, where k is the greatest common divisor of {n, r,s}.3.1.9. (!) Prove that every maximal matching in a graph G has at least a'(G)/2 edges.
- 1.2.10. (-) Prove or disprove: a) Every Eulerian bipartite graph has an even number of edges. b) Every Eulerian simple graph with an even number of vertices has an even num- ber of edges.9 / prove that: Let G be agraph with n vertices, ny 2 و the h G has at Least two vertices which are not cut vertices.1.2.6. (-) In the graph below (the paw), find all the maximal paths, maximal cliques, and maximal independent sets. Also find all the maximum paths, maximum cliques, and maximum independent sets.
- Which gra5. Let A = {1, 2, 3}. Let G be a digraph and V(G) = pow(A). E(G) = {(u, v)|u < v}, where uc v means u is a proper subset (or strict subset) of v. a) Draw the graph G. b) Let v be the vertex corresponding to the set {2}. Find all elements of G reachable from v. c) Find two different topological sorts of G.8. If the graph of h(x) goes through the points A(-16,4), B(-4,0), C(0,-4) and D(4,16), then how many of the following statements are true for the graph of h(x) ? Point A will be mapped onto A'(-16,2). II. Point B will be an invariant point. III. Point C will be an invariant point. IV. Point D will be mapped onto D' (2,16) I. А. one B. two C. three D. four ZEBRA Mild Ink M IL DLI NER. -V5 А. V5 В. 5 2/5 С. - 2/5 D. Tiw doidw 10. If cos0 = V3 sin O then the exact value of tan 20 is: A. -V3 В. 2/3 D. 3 С.