Handshaking Theorem Solved Examples
Example 1. In a party, each person shakes hands with every other person exactly once. If there were a total of \(28\) handshakes, how many people attended the party?
Solution:
Let's assume there were '\(n\)' people at the party.
Using the Handshaking Theorem, we know that the sum of the degrees of all vertices (people) is twice the number of edges (handshakes).
The sum of degrees is \(n(n - 1)\), as each person shakes hands with \((n - 1)\) others.
So, we have \(n(n - 1) = 2 \times 28\).
Solving for '\(n\)', we get \(n = 8\).
Therefore, \(8\) people attended the party.
Example 2. In a social network, if the sum of degrees of all users is \(90\), how many connections (edges) are there in the network?
Solution:
Let '\(E\)' be the number of edges.
According to the Handshaking Theorem, the sum of degrees equals \(2 \times E\).
So, we have \(90 = 2 \times E\), which implies \(E = 45\).
There are \(45\) connections in the social network.
Example 3. A sports tournament has \(12\) teams, and each team plays against every other team exactly once. How many total matches were played?
Solution:
If each team plays against every other team, we can consider each match as an edge connecting two vertices (teams).
Using the Handshaking Theorem, the sum of degrees of all vertices (teams) is twice the number of matches.
The sum of degrees is \(12(12 - 1) = 132\).
So, \(132 = 2 \times E\), where \(E\) is the number of matches.
Solving for '\(E\)', we find \(E = 66\).
There were \(66\) total matches.
Example 4. In a classroom, there are \(25\) students. If each student shakes hands with exactly \(3\) other students, how many total handshakes occur?
Solution:
Each handshake involves two students and contributes to the degree of two vertices.
So, the sum of degrees is \(2 \times 25 = 50\).
According to the Handshaking Theorem, this sum is also equal to twice the number of handshakes.
Therefore, \(50 = 2 \times E\), where \(E\) is the number of handshakes.
Solving for '\(E\)', we get \(E = 25\).
There were \(25\) total handshakes.
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