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Answer the following question Let G be a graph of size 18 and let each vertex...
Exercise 5. Let G be a graph in which every vertex has degree at least m. Prove that there is a simple path (i.e. no repeated vertices) in G of length m.
Let G -(V, E) be a graph. The complementary graph G of G has vertex set V. Two vertices are adjacent in G if and only if they are not adjacent in G. (a) For each of the following graphs, describe its complementary graph: (i) Km,.ni (i) W Are the resulting graphs connected? Justify your answers. (b) Describe the graph GUG. (c) If G is a simple graph with 15 edges and G has 13 edges, how many vertices does...
3. Let G be an undirected graph in which the degree of every vertex is at least k. Show that there exist two vertices s and t with at least k edge-disjoint paths between them. 3. Let G be an undirected graph in which the degree of every vertex is at least k. Show that there exist two vertices s and t with at least k edge-disjoint paths between them.
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.
Bounds on the number of edges in a graph. (a) Let G be an undirected graph with n vertices. Let Δ(G) be the maximum degree of any vertex in G, δ(G) be the minimum degree of any vertex in G, and m be the number of edges in G. Prove that δ(G)n2≤m≤Δ(G)n2
For a directed graph the in-degree of a vertex is the number of edges it has coming in to it, and the out- degree is the number of edges it has coming out. (a) Let G[i,j] be the adjacency matrix representation of a directed graph, write pseudocode (in the same detail as the text book) to compute the in-degree and out-degree of every vertex in the Page 1 of 2 CSC 375 Homework 3 Spring 2020 directed graph. Store results...
8. For each of the following, either draw a undirected graph satisfying the given criteria or explain why it cannot be done. Your graphs should be simple, i.e. not having any multiple edges (more than one edge between the same pair of vertices) or self-loops (edges with both ends at the same vertex). [10 points] a. A graph with 3 connected components, 11 vertices, and 10 edges. b. A graph with 4 connected components, 10 vertices, and 30 edges. c....
(Problem R-14.16, page 678 of the text) Let G be a graph whose vertices are the integers 1 through 8, and let the adjacent vertices of each vertex be given by the table below: Vertex adjacent vertices 1 (2,3,4) 2 (1,3,4) 3 (1,2,4) 4 (1,2,3,6) 5 (6,7,8) 6 (4,5,7) 7 (5,6,8) 8 (5,7) Assume that, in a traversal of G, the adjacent vertices...
12. For each of the following collection of properties, draw one graph G that satisfies them all (a) G is Bipartite and contains a vertex of degree 3 (b) G is a non-planar graph with A(G) < 3 (c) G is a tree with 5 vertices and A(G) = 4 12. For each of the following collection of properties, draw one graph G that satisfies them all (a) G is Bipartite and contains a vertex of degree 3 (b) G...
8. For each of the following, either draw a undirected graph satisfying the given criteria or explain why it cannot be done. Your graphs should be simple, i.e. not having any multiple edges (more than one edge between the same pair of vertices) or self-loops (edges with both ends at the same vertex). [10 points] a. A graph with 3 connected components, 11 vertices, and 10 edges. b. A graph with 4 connected components, 10 vertices, and 30 edges. c....