1.2.17. (!) Let G,, be the graph whose vertices are the permutations of (1,..., n}, with two permutations a₁, ..., a,, and b₁, ..., b, adjacent if they differ by interchanging a pair of adjacent entries (G3 shown below). Prove that G,, is connected. 132 123 213 312 321 231

Compute the betweenness centrality of all four vertices

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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)

I want this to be considered as a Advanced Math question pls. . Consider a graph G which is a complete bipartite graph. The graph G is defined as K(3,4), meaning it has two sets of vertices, with 3 vertices in one set and 4 in the other. Every vertex in one set is connected to every vertex in the other set, but there are no connections within a set. Calculate the number of edges in graph G. Also, determine if the graph G contains an Euler path or circuit, and justify your answer.

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...

Draw a Cayley graph for Z/7Z with generating set {2¯,5¯}.

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.

7. Let G = (V, E) be a connected graph. Use induction to show that if G contains exactly |G| = |V]. cycle, then one

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.)