Hand-Shaking Theorem Part-1

Duration: 5 min

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  1. About Course: About the Course
  2. Paper 1 | General Studies: State Wise GK
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  5. Paper 2 | Discrete Mathematical Structure: प्रतिशतता / Percentage, अनुपात, समानुपात, साझेदारी एवं मिश्रण / Ratio, Proportion, Partnership and Mixtures, औसत / Averages, आँकड़ा विश्लेषण एवं शोध सांख्यिकी / Data Analysis and Statistics in Research, क्रमचय, संचय एवं प्रायिकता / Permutation, Combination and Probability, श्रेणी / Series, आँकड़े एवं आलेखीय निरूपण / Data and Graphical Representation, आँकड़ा निर्वचन: तालिका / Data Interpretation: Tables, Digital Systems & Boolean Basics, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Set Theory, Relations, Functions, Graph Theory, Group Theory, Propositional and Predicate Logic, Probability
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  10. Paper 2 | Data Structure And Algorithms: Introduction to DS, Array, Stack, Queue, Linked List, Tree, Graphs, Hashing, Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Minimum Spanning Trees, Advanced Algorithms
  11. Paper 2 | Operating System: Introduction to OS, Process Management, CPU Scheduling, Process Synchronization, Deadlock, Threads & Process Creation, Memory Management, Virtual Memory, Disc Scheduling, File Management, Unix OS
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  13. Paper 2 | Compiler Design: Intro to Compilers, Lexical Analysis, Grammar & CFG, Syntax Analysis: Top-Down, Syntax Analysis: Bottom-Up, Semantic Analysis & SDT, Intermediate Code Gen, Code Optimization, Run Time Environment
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  15. Paper 2 | Computer Network And Information Security: Introduction to CN, Data Communication, Application Layer, Hardware basics, Cyber Threats and Malware Attacks, Network Security, DLL: Access Control, DLL: Flow Control, DLL: Error Control, DLL: Framing, Data Link Layer - Ethernet, Net Layer: IPv4 & Proto, Net Layer: IP Addressing, Net Layer:Routing Protocol, Transport Layer Services, TL: Congestion & UDP
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  17. Paper 2 | Current Trends & Technologies: कंप्यूटर की मूल बातें और हार्डवेयर / Computer Fundamentals and Hardware, पारितंत्र और जैव विविधता / Ecosystems and Biodiversity, मेमोरी, भंडारण और डेटा निरूपण / Memory, Storage and Data Representation, कंप्यूटर सॉफ्टवेयर / Computer Software, निरुपाधिक प्रतिज्ञप्तियाँ / Categorical Propositions, ऑफिस अनुप्रयोग / Office Applications, संकेत विज्ञान एवं अर्थ / Semiotics and Meaning, मल्टीमीडिया और फ़ाइल प्रारूप / Multimedia and File Formats, कंप्यूटर नेटवर्क / Computer Networks, संचार की बाधाएँ / Barriers to Communication, उच्च शिक्षा संस्थानों के प्रकार / Types of Higher Education Institutions, इंटरनेट और वर्ल्ड वाइड वेब / Internet and World Wide Web, ई-मेल और ऑनलाइन संचार / Email and Online Communication, शिक्षण विधियाँ / Methods of Teaching, साइबर सुरक्षा / Cyber Security, डिजिटल नैतिकता, बौद्धिक संपदा और समाज / Digital Ethics, IPR and Society, शोध-प्रबंध एवं लेख लेखन / Thesis and Article Writing, डिजिटल एवं नव माध्यम / Digital and New Media, क्लाउड और वितरित कंप्यूटिंग / Cloud and Distributed Computing, उभरती प्रौद्योगिकियाँ / Emerging Technologies, प्राकृतिक प्रकोप और आपदा प्रबंधन / Natural Hazards and Disaster Management, शिक्षा और शासन में ICT / ICT in Education and Governance, Emerging Tech Trends, Approaches to AI, Search Algorithms, Game Playing, Knowledge Representation, Planning, Multi Agent Systems, Fuzzy Sets, Natural Language Processing, Artificial Neural Networks, Genetic Algorithms, Artificial Intelligence, Data Warehousing & Mining, Electronic Data Interchange and E-Commerce, Terms & Abbreviations
  18. Paper 2 | Full Length Mock Tests:
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

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

  1. 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. 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.

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