Let Kn denote the simple graph on n vertices. (a) Let v be some vertex of...

Let K_n denote the simple graph on n vertices.
(a) Let v be some vertex of K_n and consider K _{n −
v}, the graph obtained by deleting
v. Prove that K _{n − v} is isomorphic to K n−1 .
(b) Use mathematical induction on n to prove the following statement:
K _n , the complete graph on n vertices, has n(n-1)/2
edges

Solutions

Expert Solution

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

(a) What is the maximum degree of a vertex in a simple graph with n vertices?...

(a) What is the maximum degree of a vertex in a simple graph with n vertices? (b) What is the maximum number of edges in a simple graph of n vertices? (c) Given a natural number n, does there exist a simple graph with n vertices and the maximum number of edges?

For any n ≥ 1 let Kn,n be the complete bipartite graph (V, E) where V...

For any n ≥ 1 let Kn,n be the complete bipartite graph (V, E) where V = {xi : 1 ≤ i ≤ n} ∪ {yi : 1 ≤ i ≤ n} E = {{xi , yj} : 1 ≤ i ≤ n, 1 ≤ j ≤ n} (a) Prove that Kn,n is connected for all n ≤ 1. (b) For any n ≥ 3 find two subsets of edges E 0 ⊆ E and E 00 ⊆ E such...

Let G be a simple undirected graph with n vertices where n is an even number....

Let G be a simple undirected graph with n vertices where n is an even number. Prove that G contains a triangle if it has at least (n^2 / 4) + 1 edges using mathematical induction.

You are given a directed graph G(V,E) with n vertices and m edges. Let S be...

You are given a directed graph G(V,E) with n vertices and m edges. Let S be the subset of vertices in G that are able to reach some cycle in G. Design an O(n + m) time algorithm to compute the set S. You can assume that G is given to you in the adjacency-list representation.

let G be a simple graph. show that the relation R on the set of vertices...

let G be a simple graph. show that the relation R on the set of vertices of G such that URV if and only if there is an edge associated with (u,v) is a symmetric irreflexive relation on G

Let Q:= {1,2,...,q}. Let G be a graph with the elements of Q^n as vertices and...

Let Q:= {1,2,...,q}. Let G be a graph with the elements of Q^n as vertices and an edge between (a1,a2,...,an) and (b1,b2,...bn) if and only if ai is not equal to bi for exactly one value of i. Show that G is Hamiltonian.

Given an undirected graph G = (V,E), consisting of n vertices and m edges, with each...

Given an undirected graph G = (V,E), consisting of n vertices and m edges, with each edge labeled from the set {0,1}. Describe and analyze the worst-case time complexity of an efficient algorithm to find any cycle consisting of edges whose labels alternate 0,1.

An m × n grid graph has m rows of n vertices with vertices closest to...

An m × n grid graph has m rows of n vertices with vertices closest to each other connected by an edge. Find the greatest length of any path in such a graph, and provide a brief explanation as to why it is maximum. You can assume m, n ≥ 2. Please provide an explanation without using Hamilton Graph Theory.

Let G be a bipartite graph with 107 left vertices and 20 right vertices. Two vertices...

Let G be a bipartite graph with 107 left vertices and 20 right vertices. Two vertices u, v are called twins if the set of neighbors of u equals the set of neighbors of v (triplets, quadruplets etc are defined similarly). Show that G has twins. Bonus: Show that G has triplets. What about quadruplets, etc.?

Use the fact that every planar graph with fewer than 12 vertices has a vertex of...

Use the fact that every planar graph with fewer than 12 vertices has a vertex of degree <= 4 to prove that every planar graph with than 12 vertices can be 4-colored.

Subjects