8 that has exactly two vertices of the- Provide an example of a graph G of...
topic: graph theory Question 5. Prove that every graph with at least two vertices contains two vertices with the same degree. Then for each n 2 2 give an example of a graph with n vertices which does not have three vertices of the same degree. Question 5. Prove that every graph with at least two vertices contains two vertices with the same degree. Then for each n 2 2 give an example of a graph with n vertices which...
Prove that every graph with two or more nodes must have at least two vertices having the same degree. Determine all graphs that contain just a single pair of vertices that have exactly the same degree.
(a) Let L be a minimum edge-cut in a connected graph G with at least two vertices. Prove that G − L has exactly two components. (b) Let G an eulerian graph. Prove that λ(G) is even.
8. This question has two parts. (i) Let G be a graph with minimum vertex degree 8(G) > k. Then prove that G has a path of length at least k. (ii) Let G be a graph of order n. If S(G) > nz?, then prove that G is connected.
Let G be a graph with n vertices and n edges. (a) Show that G has a cycle. (b) Use part (a) to prove that if G has n vertices, k components, and n − k + 1 edges, then G has a cycle.
Please write your answer clearly and easy to read. Please only answer the ones you can. I will upvote all the submitted answers. Question 5. Prove by contradiction that every circuit of length at least 3 contains a cycle Question 6. Prove or disprove: There exists a connected graph of order 6 in which the distance between any two vertices is even Question 7. Prove formally: If a graph G has the property that every edge in G joins a...
49.12. Let G be a graph with n 2 2 vertices. a. Prove that if G has at least ("21) +1 edges, then G is connected. b. Show that the result in (a) is best possible; that is, for each n 2 2, prove there is a graph with ("2) edges that is not connected. 49.12. Let G be a graph with n 2 2 vertices. a. Prove that if G has at least ("21) +1 edges, then G is...
Let G be a connected graph with m 2 vertices of odd degree. Prove that once is m/2. Let G be a connected graph with m 2 vertices of odd degree. Prove that once is m/2.
1. [10 marks) Suppose a connected graph G has 10 vertices and 11 edges such that A(G) = 4 and 8(G) = 2. Let nd denote the number of vertices of degree d in G. (i) List all the possible triples (n2, N3, n4). (ii) For each triple (n2, n3, nd) in part (i), draw two non-isomorphic graphs G with n2 vertices of degree 2, në vertices of degree 3 and n4 vertices of degree 4. You need to explain...
Please answer the question and write legibly (3) Prove that for a bipartite graph G on n vertices, we have a(G)- n/2 if and only if G has a perfect matching. (Recall that α(G) is the maximum size among the independent subsets of G.) (3) Prove that for a bipartite graph G on n vertices, we have a(G)- n/2 if and only if G has a perfect matching. (Recall that α(G) is the maximum size among the independent subsets of...