Question: (a) Let G be an undirected graph with n vertices. If G is isomorphic to its own complement , how many edges must G have?

(a) Let G be an undirected graph with n vertices. If G is isomorphic to its own complement , 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 k ∈ Z+.

Step by Step Solution

★★★★★

3.47 Rating (163 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

a Let e 1 be the number of edges in G and e 2 the number in For any loop free ... View full answer

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Document Format (1 attachment)

954-M-L-A-L-S (8112).docx

120 KBs Word File

Students Have Also Explored These Related Linear Algebra Questions!

Let G be an undirected graph with vertex set V. Define the relation R on V by v R w if v = w or if there is a path from v to w (or from w to v since G is undirected), (a) Prove that is an equivalence...

(a) Let G = (V, E) be an undirected graph, with G1 = (V1, E1) a subgraph of G. Under what condition(s) is G1 not an induced subgraph of G? (b) For the graph G in Fig. 11.27(a), find a subgraph that...

Let G = (V, E) be an undirected graph with subset I of V an independent set. For each a I and each Hamilton cycle C for G, there will be deg (a) - 2 edges in E that are incident with a and not in C....

Let G be an undirected graph with n vertices and m edges. Describe an O(n + m)-time algorithm to determine whether G contains at least two cycles. Let G be an undirected graph with n vertices and m...

3. (10 points) In a directed graph, strongly connected component is defined as a subset of vertices C E V that are mutually reachable from each other, which means that for any pair of vertices v, w E...

Bounds on the number of edges in a graph. (a) Let G be an undirected graph with n vertices. Let (G) be the maximum degree of any vertex in G, (G) be the minimum degree of any vertex in G, and m be...

Let G be an undirected graph with n vertices and m edges. Argue that (a) If G is connected, then m n 1. (b) If G is a tree, then m = n 1.

Let G be an undirected graph with n vertices and m edges. Argue that (a) If G is connected, then m greaterthanorequalto n - 1. (b) If G is a tree, then m = n - 1

Let G be an undirected graph with n vertices and m edges. Argue that (a) If G is connected, then m n 1. (b) If G is a tree, then m = n 1.

A water trough has an isosceles triangular cross section that is 40 m across the top and 15 m deep. The trough is 2 m long. It is being filled at a rate of 0.4 m3 / minute. How fast does the water...

draw and explain generic block diagram of an IoT device. explain in what circumstances you will prefer MCU based board over SBC based platform for prototyping your IoT application.

5.23 Refer to Problem 5.22. What is the probability that for every five children who participated in the survey exactly five children dislike eating vegetables?

Mrs. Laval was a patient of Dr. Leopold, a practicing psy chiatrist. Dr. Leopold shared an office with two associ ates practicing in the same field. No receptionist or other employee attended the...

Express p (x) = x3 as a Taylor polynomial about a = t. In calculus, you learn that a Taylor polynomial of degree n about a is a polynomial of the form p(x) = a0 + a1(x - a) + a2(x - a)2 + + an(x -...

Let B, C, and V be bases for a finite-dimensional vector space V. Prove that PDC PCB = PDB

Let V be an n-dimensional vector space with basis B = {v1 , . . . , vn}. Let P be an invertible n X n matrix and set ui = p1iv1 + + pni vN for i = 1 , . . . , n. Prove that C = {u1 , . . . , un}...

Use the following data for problems 25 - 27 Paradise Real-estate developer is considering building a condominium project. Five different sizes are being considered. The probabilities reflect the...

Source: Naulor WW 29. The Empirical Rule The Stanford-Binet Intelligence Quotient (IQ) measures intelligence. IQ scores have a bell shaped distribution with a mean of 100 and a standard devin of 15....

Show that any one-asymmetric error correcting code of length n can have a maximum of 2^n/(1 +n/2) codewords.