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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- 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? ...1The 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...1.2.20. (!) Let u be a cut-vertex of a simple graph G. Prove that G - v is connected. ע
- 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.Prove that If a connected planar simple graph has e edges and v vertices with v ≥ 3 and no circuits of length three, then e ≤ 2v − 4. (Show work)18. Let G be a graph with n vertices and exactly n-1 edges . Prove that G has either a vertex of degree 1 or an isolated vertex.
- 3.1.2. (-) Determine the minimum size of a maximal matching in the cycle C,.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.
- Let P₁ and P₂ be two paths of maximum length in a connected graph G. Prove that P₁ and P2 have a common vertex.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.4a Let n 2 4. WVhat is the maximum possible number of edges in a graph with n vertices and n - 2 connected components? Prove your answer. 4b How many different undirected graphs can be formed with vertex set V = {1,2,3, 4}? 2}) and (V, {2 – 3}) as two different (The vertices are distinguishable, so we count (V,{1 graphs, for example.)