Hand-Shaking Theorem Part-1
Duration: 5 min
This video lesson is available to enrolled students.
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This educational video provides a detailed explanation of the Hand-shaking theorem in graph theory. The instructor defines the theorem, which states that the sum of the degrees of all vertices in a graph is equal to twice the number of edges. He then derives a significant corollary: the number of vertices with odd degrees in any graph must always be an even number. The lesson is supported by a visual example featuring a graph with vertices labeled 'a' through 'j' and a corresponding table listing the degree of each vertex.
Chapters
0:00 – 2:00 00:00-02:00
The video begins with the definition of the Hand-shaking theorem displayed on the screen: 'Since each edge contribute two degree in the graph, the sum of the degree of all vertices in G is twice the number of edges in g.' The instructor writes the mathematical formula $\sum_{i=1}^{n} d(vi) = 2|E|$ on the whiteboard area. He explains the logic that every edge connects two vertices, contributing 1 to each, totaling 2 per edge. He underlines the summation symbol and the term $2|E|$ to emphasize the equality. He then points to the graph diagram on the left and the 'Vertex | Degree' table on the right, preparing to verify the theorem with the specific data provided.
2:00 – 5:00 02:00-05:00
The instructor introduces a new theorem: 'The number of vertices of odd degree in a graph is always even.' He writes the equation $\sum_{i=1}^{n} d(vi) = \sum_{even} d(vi) + \sum_{odd} d(vi)$ to break down the total sum. He explains that the total sum is even (from the Hand-shaking theorem) and the sum of even degrees is inherently even. Therefore, the sum of odd degrees must also be even, which is only possible if there is an even number of odd-degree vertices. To prove this, he circles the vertices with odd degrees in the table: 'a' (1), 'b' (1), 'c' (1), 'f' (1), 'g' (3), and 'h' (3). He counts these circled entries, confirming there are exactly 6 vertices of odd degree, which satisfies the theorem.
The lecture effectively connects the abstract theorem to a concrete example. By calculating the sum of degrees (1+1+1+4+4+1+3+3+2+2 = 22) and comparing it to twice the number of edges (11 edges * 2 = 22), the instructor validates the Hand-shaking theorem. The second part of the lecture logically deduces the parity of odd-degree vertices, a fundamental result in graph theory often used to prove the existence of Eulerian paths or cycles. The visual aids, including the red checkmarks and circles, help students track the specific vertices being discussed.