Number Of Simple Graph With n Vertices

Duration: 5 min

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Module outline

  1. Discrete Mathematics: Permutation, Combination and Probability / क्रमचय, संचय एवं प्रायिकता, Set Theory, Relations, Functions, Propositional and Predicate Logic, Graph Theory, Group Theory
  2. Digital Logic: Digital Systems & Boolean Basics, Boolean Expression, Boolean Minimization, Logic Gates & Hardware, Combinational Circuit, Sequential Circuits, Number System, Number Representation, Floating Point Rep
  3. Computer Organization and Architecture: Basics of COA, Programming the Basic Computer, Cache Memory Organization, Input Output Organisation, Instr Formats & Modes, Register Transfer and Microoperations, Control Unit Design, Pipelining, Multiprocessors
  4. Programming: Language Design, C Fundamentals, Control Flow, Functions, Arrays & Pointers, Storage Classes, Structures & Enums, DMA, Macros, Scoping & File Handling, OOPS with C++
  5. Data Structures: Introduction to DS, Array, Linked List, Stack, Queue, Tree, Graphs, Hashing
  6. Algorithms: Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Dynamic Programming, Minimum Spanning Trees, Shortest Path Algos, Advanced Algorithms
  7. Theory of Computation: Grammar, Regular Language Properties, Moore & Mealy Machines, Turing Machines
  8. 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, Introduction to TOC, Deterministic FA (DFA), Non-Deterministic FA, Regular Expressions, Pushdown Automata & CFG, Complexity Theory
  9. Operating Systems: Introduction to OS, Process Management, CPU Scheduling, Process Synchronization, Threads & Process Creation, Deadlock, Memory Management, Virtual Memory, Disc Scheduling, File Management, Unix OS, Linux OS, Security
  10. Database Management Systems: Basics of DBMS, ER Diagram, Relational Model & Functional Dependencies, Keys & Integrity Constraints, Normalization (1NF - BCNF), Decomposition Properties & 4NF, File Organization & Indexing, Relational Algebra, SQL, Relational Calculus, Transaction Management, Concurrency Control, ORDBMS, NoSQL
  11. Computer Networks: Introduction to CN, Data Communication, 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, Application Layer, Hardware basics, Network Security
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

The video lecture addresses the combinatorial problem of determining the total number of distinct simple graphs that can be formed with a given number of vertices, denoted as 'n'. The instructor begins by breaking down the problem for small values of n, specifically focusing on the case where n=3. He systematically enumerates all possible simple graphs by categorizing them according to the number of edges they contain, ranging from zero edges to the maximum possible edges. Through this enumeration, he demonstrates that for three vertices, there are exactly 8 possible simple graphs. The lecture then transitions to deriving a general formula for any number of vertices 'n' by considering the total number of potential edges in a complete graph. Finally, the instructor introduces a related but distinct problem: finding the number of simple graphs with 'n' vertices and a specific number of edges 'e'.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The instructor poses the question "Number of simple graph possible with n vertices?" and begins a case-by-case analysis for n=3. He draws vertical columns to separate cases based on the number of edges, labeled |E|=0, |E|=1, |E|=2, and |E|=3. For |E|=0, he draws three isolated vertices. For |E|=1, he draws three distinct graphs, each with a single edge connecting a different pair of vertices. For |E|=2, he draws three graphs, each missing a different edge from the complete triangle. For |E|=3, he draws the complete graph K3 (a triangle). He sums the counts (1+3+3+1) to arrive at a total of 8 possible graphs for n=3.

  2. 2:00 – 4:37 02:00-04:37

    The instructor generalizes the previous result. He explains that the total number of graphs is 2 raised to the power of the number of possible edges. He writes the formula for the number of possible edges in a simple graph with n vertices as n(n-1)/2. Consequently, he writes the final formula for the total number of simple graphs as 2^(n(n-1)/2). He then moves to a new slide or question text: "Number of simple graph possible with n vertices and e edges". He underlines the terms "n vertices" and "e edges" to emphasize the constraints of this new problem, preparing to discuss combinations.

The lesson progresses from a concrete enumeration example to an abstract formula. By analyzing the n=3 case, the instructor illustrates that the number of graphs corresponds to the number of subsets of the set of all possible edges. Since each possible edge can either exist or not exist (2 choices), and there are n(n-1)/2 possible edges, the total count is 2^(n(n-1)/2). The video concludes by pivoting to a more constrained counting problem where the number of edges 'e' is fixed, shifting the focus from subsets of all possible edges to combinations of exactly 'e' edges chosen from the total possible edges.

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