1. Solve the following: (a) Let G be an undirected graph with n vertices. If G is isomorphic to its own complement G,how many edges must G have? (Such a graph is called self-complementary.)(b) Find an example of a self-complementary graph on four vertices and one on five vertices.(c) If G is a self-complementary graph on n vertices, where n > 1, prove that n = 4k or n = 4k+1,for some keZ+.

Step 1:(a) Number of Edges in a Self-Complementary GraphA graph G with n vertices is called…

Answered: (a) Let G be an undirected graph with n…

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter5: Exponential And Logarithmic Functions

Section5.3: Logarithmic Functions And Their Graphs

Problem 137E

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A simple graph that is isomorphic to its complement is self-complementary. (i) Prove that, if G is self-complementary, then G has 4k or 4k+1 vertices, where k is an integer. (ii) Find all self-complementary graphs with 4 and 5 vertices. (iii) Find a self-complementary graph with 8 vertices.
At a cupcake stall at the Greenwich market, the owner is selling six different types of cupcakes. The owner sells them in boxes where each box contains exactly two cupcakes, each of different type. Each type of cupcake is used in combination with at least three others. (a) Let G be a graph where each vertex represents one of the cupcakes and where an edge joins two vertices whenever two cupcakes are used in the same box. Let v be an arbitrary vertex in G. What is the smallest degree v can have? Justify your answer. ' ^ (b) Show that there are three gift boxes which between them have all six types of cupcakes.
Let G be a graph with 10 vertices and 5 components. If a vertex is removed from G, the number of components in the resultant graph must lie down between and 10 and 4 9 and 4 10 and 3 9 and 5
3. (a) Is it possible to have a 4-regular graph with 15 vertices? If no, explain why. If yes, construct such a graph. (b) The degree of every vertex of a graph G is one of three consecutive integers. If, for each of the three consecutive integers r, the graph G contains exactly r vertices of degree r, prove that two-thirds of the vertices of G have odd degree.
Let G be a graph of order n that is isomorphic to its complement G. How many edges does G have? Explain your answer. If a graph G has n vertices, all of which but one have odd degree, how many vertices of odd degree are there in G, the complement of G? Prove your answer.
2. Give an example of a graph with at least four vertices, or prove that none exists, such that: (a) There are no vertices of odd degree. (b) There are no vertices of even degree. (c) There is exactly one vertex of odd degree. (d) There is exactly one vertex of even degree. (e) There are exactly two vertices of odd degree.
Prove that the two graphs below are isomorphic. (c) Figure 4: Two undirected graphs. Each graph has 6 vertices. The ver- tices in the first graph are arranged in two rows and 3 columns. From left to right, the vertices in the top row are 1, 2, and 3. From left to right, the vertices in the bottom row are 6, 5, and 4. Undirected edges, line segments, are between the following vertices: 1 and 2; 2 and 3; 1 and 5; 2 and 5; 5 and 3; 2 and 4; 3 and 6; 6 and 5; and 5 and 4. The vertices in the second graph are a through f. Vertices d, a, and c, are vertically inline. Vertices e, f, and b, are horizontally to the right of vertices d, a, and c, respectively. Undirected edges, line segments, are between the following vertices: a and d; a and c; a and e; a and b; d and b; a and f; e and f; c and f; and b and f.
When n = 3, there are nonisomorphic simple graphs.
Let Vn be the set of connected graphs having n edges, vertex set [n], and exactly one cycle. Form a graph Gn whose vertex set is Vn. Include {gn, hn} as an edge of Gn if and only if gn and hn differ by two edges, i.e. you can obtain one from the other by moving a single edge. Tell us anything you can about the graph Gn. For example, (a) How many vertices does it have? (b) Is it regular (i.e. all vertices the same degree)? (c) Is it connected? (d) What is its diameter?
(c) Prove that the two graphs below are isomorphic. Figure 4: Two undirected graphs. Each graph has 6 vertices. The ver- tices in the first graph are arranged in two rows and 3 columns. From left to right, the vertices in the top row are 1, 2, and 3. From left to right, the vertices in the bottom row are 6, 5, and 4. Undirected edges, line segments, are between the following vertices: 1 and 2; 2 and 3; 1 and 5; 2 and 5; 5 and 3; 2 and 4; 3 and 6; 6 and 5; and 5 and 4. The vertices in the second graph are a through f. Vertices d, a, and c, are vertically inline. Vertices e, f, and b, are horizontally to the right of vertices d, a, and c, respectively. Undirected edges, line segments, are between the following vertices: a and d; a and c; a and e; a and b; d and b; a and f; e and f; c and f; and b and f. (d) Show that the pair of graphs are not isomorphic by showing that there is a property that is preserved under isomorphism which one graph has and the other does not. Figure 5: Two…
Show that if G is a simple graph with n vertices (where n is a positive integer) and each vertex has degree greater than or equal to "1, then the diameter of G is 2 or less. If G is a (not necessarily simple) graph with n vertices n-1 2 where each vertex has degree greater than or equal to ", is the diameter of G necessarily 2 or less? Either prove that the answer to this question is "yes" or give a counterexample.

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