Number Of Simple Graph With n Vertices e Edges

Duration: 3 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.

This educational video segment addresses a fundamental problem in graph theory: determining the total number of distinct simple graphs that can be formed with a fixed number of vertices (n) and a fixed number of edges (e). The instructor begins by defining the parameters for a specific example, setting the number of vertices to 4. He systematically derives the range of possible edges, identifying the minimum as zero and calculating the maximum using the standard formula for a complete graph. The core of the lesson involves applying combinatorial mathematics, specifically the combination formula, to select the required number of edges from the total pool of possible connections.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The instructor writes the question 'Number of simple graph possible with n vertices and e edges' on the screen. He establishes the example parameters by writing |V| = 4. He explains that the minimum number of edges (E_min) is 0. He then derives the formula for the maximum number of edges (E_max) in a simple graph as n(n-1)/2. Substituting n=4, he calculates 4 * 3 / 2 = 6. He introduces the combination notation ^nC_r and its factorial formula n! / (r! (n-r)!), setting up the framework to solve for the number of graphs.

  2. 2:00 – 2:47 02:00-02:47

    The instructor proceeds to solve the specific problem for e=2 edges. He writes the combination expression ^6C_2, indicating the selection of 2 edges from the 6 possible edges calculated previously. He expands the factorials explicitly as 6! / (2! 4!). He simplifies the fraction by canceling out 4!, leaving (6 * 5) / 2. He performs the final arithmetic to get 15 and circles the result, concluding that there are 15 possible simple graphs with 4 vertices and 2 edges.

The lecture demonstrates a method for counting simple graphs by treating the set of all possible edges as a pool from which a specific subset must be chosen. By first calculating the total number of potential edges for a given number of vertices using the formula n(n-1)/2, the problem is reduced to a standard combinatorial selection. The instructor illustrates this by calculating the number of ways to choose 2 edges from the 6 available edges in a 4-vertex graph, resulting in 15 distinct graphs. This approach highlights the relationship between graph structure and combinatorial mathematics.

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