oneamp said:Homework Statement
Prove that if a graph has > (n-1)(n-2) /2 edges, it is connected.
Homework Equations
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The Attempt at a Solution
I've drawn several examples and made tables, and I can see that the graph is indeed connected if it has more edges than [(n-1)(n-2)]/2. But what I cannot do so far is prove it. How can I start doing this proof?
Thanks
The main purpose of a connection proof in Graph Theory is to demonstrate that there is a path between two given vertices in a graph. This is important because it helps to establish relationships between different elements in a graph and can aid in solving complex problems.
A connection proof in Graph Theory typically consists of three main parts: the statement of the connection, the proof of the connection, and the conclusion. The statement of the connection clearly defines the two vertices that are being connected, the proof outlines the steps taken to establish the connection, and the conclusion summarizes the result.
Some common techniques used in connection proofs in Graph Theory include induction, contradiction, and direct proof. Induction involves proving a connection for a base case and then showing that it holds true for all subsequent cases. Contradiction involves assuming that there is no connection and then showing that this assumption leads to a contradiction. Direct proof involves using logical reasoning and previous knowledge to establish a connection.
Yes, a connection proof can be used to determine if a graph is connected. If a connection proof can be established between any two vertices in a graph, then the graph is considered to be connected. However, if a connection proof cannot be established between any two vertices, then the graph is considered to be disconnected.
Yes, there are many real-life applications of connection proofs in Graph Theory. For example, connection proofs are commonly used in computer networks to ensure that all computers are connected and can communicate with each other. They are also used in transportation networks to determine the most efficient routes between two locations. In addition, connection proofs are used in social networks to analyze the relationships between individuals and identify potential connections.