Degree of Vertex, Isolated & Pendant Vertex

Duration: 4 min

This video lesson is available to enrolled students.

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  3. Paper 1 | General Hindi: भाषा, लिपि, ध्वनि, वर्ण, अक्षर, स्वर, व्यंजन, मात्रा और संयुक्ताक्षर, वर्तनी, वर्ण-विन्यास, अनुस्वार, अनुनासिक, चंद्रबिंदु और ‘र’ के रूप, शब्द, पद और शब्द-भेद, संज्ञा और सर्वनाम, विशेषण और क्रिया, लिंग, वचन, पुरुष और कारक, काल, वाच्य, वाक्य और वाक्य-भेद, वर्णमाला, समास, संधि, Multi-Topic Content (Cross-Concept Material), हिन्दी की ध्वनि और स्वनिम व्यवस्था, प्राचीन और मध्यकालीन भारतीय आर्यभाषाएँ, काव्य के लक्षण, हेतु और प्रयोजन, हिन्दी की रूप रचना, हिन्दी शब्द रचना, हिन्दी साहित्येतिहास दर्शन, भक्तिकालीन काव्यधाराएँ, भारतीय काव्यशास्त्र की परंपरा, भारतीय नवजागरण और स्वाधीनता आंदोलन, अन्य गद्य विधाएँ एवं प्रवासी साहित्य, निर्धारित उपन्यासों का तुलनात्मक अध्ययन, गोदान, हिन्दी कहानी का विकास, स्वातंत्र्योत्तर और नयी कहानी, हिन्दी निबंध का विकास, शुक्लोत्तर निबंध, द्विवेदी और शुक्ल युग के निबंध, हिन्दी की संवैधानिक स्थिति, आधुनिक भारतीय आर्यभाषाएँ और हिन्दी का उद्भव, हिन्दी भाषा प्रयोग के विविध रूप, रीतिकाल, जीवनी, द्विवेदी युग और छायावाद, रीति, वक्रोक्ति और औचित्य सिद्धान्त, शब्दशक्ति और ध्वनि सिद्धान्त, रस सिद्धान्त और साधारणीकरण, उत्तर आधुनिकतावाद, लोहिया दर्शन, हिन्दी उपन्यास का विकास, हिन्दी नवजागरण, अंबेडकर दर्शन, आदिकाल, प्रयोगवाद और नयी कविता, छायावादी काव्य: प्रसाद और निराला, प्रगतिवाद से समकालीन कविता, निर्गुण भक्ति काव्य, द्विवेदीयुगीन काव्य, रीतिकालीन काव्य, आरंभिक हिन्दी उपन्यास, स्वातंत्र्योत्तर उपन्यास, आत्मकथा, स्वाधीनता-पूर्व प्रेमचंदोत्तर उपन्यास, निर्धारित कहानियों का तुलनात्मक अध्ययन, साठोत्तरी नाटक, प्रसाद और प्रसादोत्तर युग के नाटक, भारतेन्दु युगीन नाटक, स्वातंत्र्योत्तर नाटक, भारतेन्दु युगीन निबंध, संस्मरण और रेखाचित्र, व्यंग्य, एक साहित्यिक की डायरी, संस्कृति के चार अध्याय, सगुण भक्ति काव्य, हिन्दी नाटक, एकांकी और रंगमंच, आधुनिक गद्य और हिन्दी नवजागरण, अलंकार सिद्धान्त, हिन्दी आलोचना का विकास, भक्ति आंदोलन, कल्पना, बिम्ब, प्रतीक, मिथक और फंतासी, नयी कविता का विस्तार और साठोत्तरी कविता, उत्तर-छायावादी और प्रगतिवादी काव्य, काव्यगुण और काव्यदोष, प्लेटो, अरस्तू और पाश्चात्य परंपरा, मार्क्सवाद, स्वच्छन्दतावादी काव्यशास्त्र, आधुनिक पाश्चात्य आलोचना, भाषा और भाषा परिवार, देवनागरी लिपि
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  23. UPPSC Live Class Recordings: UPPSC Polytechnic Lecturer 2025 (CS)
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AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This educational video provides a detailed introduction to graph theory concepts, specifically focusing on the Degree of a Vertex and the Hand-shaking theorem. The instructor begins by defining the degree of a vertex in an undirected graph as the count of edges connected to that specific vertex. He uses a visual example with vertices labeled 'a' through 'j' to demonstrate the calculation process. The lecture further classifies vertices based on their degrees, introducing terms like isolated vertex and pendant vertex. Finally, the session culminates in the Hand-shaking theorem, which establishes a mathematical relationship between the sum of all vertex degrees and the total number of edges in the graph.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The instructor introduces the core definition: Degree of a Vertex: The degree of a vertex in an undirected graph is the number of edges associated with it, denoted by deg(vi). He underlines key phrases on the slide to emphasize the definition. He then analyzes a specific graph diagram containing vertices 'a' through 'j'. A table is displayed next to the graph, listing the calculated degree for each vertex (e.g., vertex 'd' has a degree of 4, vertex 'e' has a degree of 4). He circles vertices 'a', 'b', and 'c' to identify them as having a degree of 1. He also writes deg(k) = 0 to explain the concept of an isolated vertex, which has no edges. The slide explicitly defines a Pendant vertex as a vertex with degree one.

  2. 2:00 – 3:40 02:00-03:40

    The lecture transitions to the Hand-shaking theorem. The slide text states: 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 explains that because every edge connects two vertices, it contributes exactly two to the total degree sum. The mathematical formula sum(d(vi)) = 2|E| is shown on the screen. This theorem provides a method to verify the consistency of a graph's structure. The instructor likely uses the previous example graph to demonstrate that the sum of the degrees equals twice the number of edges, confirming the theorem.

The video builds a coherent understanding of graph connectivity. It starts with the local property of a single vertex (degree), moves to classifying specific types of vertices (pendant, isolated), and concludes with a global property (Hand-shaking theorem). This progression allows students to understand how local connections sum up to define the entire graph's structure. The visual aids, including the graph diagram and the degree table, are crucial for reinforcing the abstract definitions with concrete examples. The theorem serves as a fundamental check for graph validity.

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