7. We define a forest to be a graph with no cycles. a. Explain why this is a good name. That is, explain why a forest is a union of trees. b. Suppose F is a forest consisting of m trees and v vertices. How many edges does F have? Explain. c. Prove that any graph G with v vertices and e edges that satisfies v

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Q: 3. Show that the two drawings represent the same graph by labeling the vertices and edges of the…

A: Give figure is

Q: Descrete math. Graph theory

A:

Q: 1. Suppose G is a planar graph with four faces of lengths 3, 4, 5, 6. How many edges are there in G?

A: Answer :-  a). Suppose G is a planar graph with faces of length 3, 4, 5, 6. How many edges are there…

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Q: Let G be a (p, q) graph. Prove that the sum of the degrees of the vertices of G equals twice the…

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Q: Consider a bipartite graph on the set A = {1,2,3, 4, 5, 6} and B = {7,8,9, 10, 11, 12}, there is an…

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Q: Construct a connected bipartite graph that is not a tree with vertices U, V,W, X,Y. What is the edge…

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Q: 13. A) Prove the statement: If G is a bipartite graph, then it does not contain cycles of odd…

A: Yes , The converse of statement A is true . Below I provide the detail proof of this :-

Q: 12. Let G be a graph in which there is no pair of adjacent edges. What can you say about the degree…

A: See picture

Q: Let G = (V, E) be a graph (no loops, no multi-edges, undirected, as always. Let d be the smallest…

A: ANSWER:

Q: 13. Determine whether the following graph is strongly connected. b.

A:

Q: 9. Verify that Euler's formula does not hold for a graph that is not connected: count the vertices,…

A: The given question is related with graph theory. We have to verify that Euler's formula does not…

Q: 6. Which of the next sets establish a bipartition of the graph in Fig.1? a) {a, h, g, b, f} and {i,…

A: To determine which of the given sets establish a bipartition of the graph. Here we will use the…

Q: (9) Consider a graph with the following set of vertices (v1, v2, v3, v4) and set of edges…

A: In our graph, we have vertices and edges, or both are defined, and to obtain a walk we need to…

Q: 3. The complement of a graph G= (V, E) is the graph G = (V,E) where E = {{r, y} | 1, y E V(G), {r,…

A: a) Let Kn denote the complete graph of n vertices. The complement of the complete graph Kn is…

Q: Let G be a bipartite graph with bipartition (X, Y). For SCX, let E, be the set of edges in G…

A: Given below the detailed solution

Q: 1. Let F be a forest with 50 vertices and 35 edges. How many trees does have F.

A:

Q: a. Let V = (x, y, z, u, v). How many di exactly 6 edges? (Note that for graphs G₁ = (V, E₁) and G₂ =…

A: (a) We are given that the set of vertices is V=x, y, z, u, v. We are asked to find the number of…

Q: 5. Determine whether the given pair of graphs is isomorphic. u2 v2 V3 V4

A: The number of vertices in graph 1= 8 The number of vertices in graph 2 = 8  The number of vertices…

Q: 1. Let G be a complete graph with n vertices. a. How many edges does G have? Explain why. b. What is…

A: Solution of given problem as follow:

Q: The complement of a graph G is a new graph formed by removing all the edges of G and replacing them…

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Q: 4. Imagine that the diagram shown below is a map with countries labelled a-g. Is it possible to…

A: Given - Imagine that the diagram shown below is a map with countries labelled a-g.  To find - Is it…

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Q: The set of all finite connected simple graphs is countable. O true O false

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Q: Which of the following statements about a connected graph is always TRUE?         A path of…

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Q: O Let G be a such that graph with 10 verti ces 8CG) = ACG) = 2 How many edges does G have?

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Q: Suppose that G is a k-regular graph of n vertices, and m edges. The vertex connectivity of G is…

A: Sol

Q: Suppose we add two edges in a tree T to form a graph G. How many cycles can be there in G.

A: When 1 edges are added to a tree,  a cycles are created When 2 edges are added to a tree, either 2…

Q: Consider the following statements: S: There exists a connected simple graph with 6 vertices and 5…

A: We have to verify the given statements.

Q: 10) Determine if the graph contains a Hamilton path or circuit. If so, write the path or circuit. F…

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Q: If a planar graph has 18 vertices and 26 edges, how many regions are determined when you draw t…

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Q: 5. For any graph G with six vertices, then G or its complement Ġ contains a triangle.

A: The solution for the above question is as shown below.

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Q: 20 8. Compute the crossing number for each of the following graphs.

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Q: A. If a simple graph G on 30 vertices has at least 301 edges, then its chromatic number is at least…

A: Note: As per bartleby instruction when more then one question is given only one has to be answered.…

Q: a. Let V = (x, y, z, u, v). How many distinct graphs are there on the exactly 6 edges? (Note that…

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Q: If G is a forest with 30 vertices and 6 con- nected components, then how many edge does G have?

A:

Q: 2) Find, with a proof, the number of distinct graphs with vertex set [n].

A: 2) The number of distinct graphs of vertex set for given n should be found. Let us take n=3.…

Q: The following graph has a Hamilton circuit: 2 13 7 9. 11 10 14 13 12 15 17 16 O True False 00

A:

Q: 15. Determine if the graph contains an Eulerian path or circuit. If so, write the path or circuit. B…

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Q: Let A and B each be sets of N labeled vertices, and consider bipartite graphs between A and B. How…

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Q: Prove that if there is a circuit in a graph that starts and ends at a vertex v and if w is another…

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Q: Let G be a graph and let v and w be vertices in G. (a) A walk from v to w is _____. (b) A trail from…

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Q: 2. The complement of a graph G = (V, E) is the graph G = (V, E) where E = {{x,y} | x, y = V(G),…

A: please comment if you need any clarification. If you find my answer useful please put thumbs up.…