1.2.18. (!) Let G be the graph whose vertex set is the set of k-tuples with elements in (0, 1), with x adjacent to y if x and y differ in exactly two positions. Determine the number of components of G.
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- Let P₁ and P₂ be two paths of maximum length in a connected graph G. Prove that P₁ and P2 have a common vertex. Let G be a graph of order n and size strictly less than n - - 1. Prove that G is not connected.Let P₁ and P₂ be two paths of maximum length in a connected graph G. Prove that P₁ and P2 have a common vertex.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.
- 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...1.2.6. (-) 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.Show that For n > 1 let Gn be the simple graph with vertex set V(Gn) = {1,2, ., n} in which two different vertices i and j are adjacent whenever j is a multiple of i or i is a multiple of j. For what n is Gn planar? ...1
- 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.Please give me answer very fast in 5 min3. [10 marks] Let Go (Vo, Eo) and G₁ = (V1, E1) be two graphs that ⚫ have at least 2 vertices each, ⚫are disjoint (i.e., Von V₁ = 0), ⚫ and are both Eulerian. Consider connecting Go and G₁ by adding a set of new edges F, where each new edge has one end in Vo and the other end in V₁. (a) Is it possible to add a set of edges F of the form (x, y) with x € Vo and y = V₁ so that the resulting graph (VUV₁, Eo UE₁ UF) is Eulerian? (b) If so, what is the size of the smallest possible F? Prove that your answers are correct.
- Let G be a graph with n ≥ 3 vertices that has a clique of size n − 2 but no cliques of size n − 1.Prove that G has two distinct independent sets of size 2. Show your work and complete proof.i) Consider the set S = {−6, −3, 0, 3, 6}. Draw a graph G whose set of verti- ces be S and such that for i, j ∈ S, ij ∈ E(G) if ij are related to a rule that t'u you choose to apply to i and j. (ii) A graph G of order 12 has as a set of vertices c1, c2, . . . , c12 for the do- ce configurations of figure 1. A movement on said board corresponds to moving a coin to an unoccupied square using the following two rules: 1. the gold coin can move only horizontally or diagonally, 2. the silver coin can move only vertically or diagonally. Two vertices ci, cj, i̸ = j are adjacent if it is possible to move ci to cj in a single movement. a) What vertices are adjacent to c1 in G? b) Draw the subgraph induced by {c2, c6, c9, c11}