Euler Formula and its Version

Duration: 4 min

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  1. General Intelligence & Reasoning: Analogy, Non Verbal Reasoning (Spatial Aptitude) (Spatial Reasoning) (Visual Reasoning), Series (Number and Letter Series) (Numerical Relations and Reasoning), Coding Decoding, Blood Relations, Directions (Direction Test), Ranking, Order and Sequence, Logical Sequence Of Words, Syllogisms, Venn Diagram, Deductive and Inductive Reasoning (Logical Deduction and Induction) (Prepositional Reasoning), Selection Decision Table (Decision Making), Data Sufficiency, Puzzle Solving (Puzzles), Data Arrangement, Cubes & Dices, Calendar, Clock, Arithmetic, Number System, HCF LCM, Verbal Analogy (Word Based Analogy), Visual Recognition and Figure Completion, Paper Folding and Cutting, Transformation of Shapes, Clocks and Calendars, Direction Sense, Cubes and Dice, Patterns in Two-Dimensional Figures, Seating Arrangements, Analogy and Classification, Logical Deduction and Induction, Alphabet and Word Tests, Mathematical Operations, Series Completion, Simplification and Approximation, Coding and Decoding, Numerical Reasoning, Assembling and Grouping of Shapes, Three-Dimensional Patterns and Visualization, Vocabulary, Number System, Averages, Mixtures and Alligation, Simple and Compound Interest, Percentages, Algebra, Speed, Time and Distance, Seating and Arrangement Puzzles, Traverse Surveying, Order, Ranking and Sequence, Constraint Puzzles, Syllogisms and Venn Diagrams, Reading Comprehension, Critical Reasoning, Permutations and Combinations, Narrative Sequencing, Project Monitoring, Resources and Safety, Profit, Loss and Discount, Ratio and Proportion, Time and Work, Mensuration, Data Interpretation, Powers, Surds and Logarithms, Sequences and Series, Articles and Determiners, Sentence Completion, Data Sufficiency, Blood Relations, Simple Interest and Compound Interest, Mixture and Alligation, Sequence and Series (Series), Speed Time and Distance, Algebra, Profit and Loss, Powers and Exponents (Surds and Indices), Permutation and Combination
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  3. Engineering Mathematics: Set Theory, Relations, Functions, Graph Theory, Group Theory, Propositional and Predicate Logic, Linear Algebra, Calculus, Probability, Statistics, Signal Space and Optimum Detection, Probability, Statistics
  4. Digital Logic: Digital Systems & Boolean Basics, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Combinational Circuit, Sequential Circuits, Number System, Number Representation, Floating Point Rep, Number Systems and Codes, Registers and Counters, Boolean Algebra and Logic Gates, Microprocessors and Microcontrollers, Logic Families and CMOS Logic, Microprocessors, Logic Gates and Boolean Algebra, Minimization of Boolean Functions, Timing, Hazards and Testing, Logic Families, Combinational Logic Design, Arithmetic Circuits, Waveform Generators and Timer Circuits, Semiconductor Memories and Programmable Logic Devices, Combinational Logic Circuits, Latches and Flip-Flops, Finite State Machines, Sequential Circuits and State Machines, Minimization of Boolean Functions, Arithmetic Circuits, Latches and Flip-Flops, Number Systems and Codes, Registers and Counters
  5. Computer Organisation & Architecture: Basics of COA, Cache Memory Organization, Input Output Organisation, Pipelining, Instr Formats & Modes, Control Unit Design, Programming the Basic Computer, Register Transfer and Microoperations, Semiconductor Memories and Programmable Logic, Processor Organization and Instructions, Memory Hierarchy and Input/Output, Registers and Counters
  6. Programming & Data Structures: C Fundamentals, Control Flow, Functions, Arrays & Pointers, Storage Classes, Structures & Enums, DMA, Macros, Scoping & File Handling, Introduction to DS, Array, Stack, Queue, Linked List, Tree, Graphs, Hashing, HTML Basics, OOPS with C++, Java, Object Oriented Technology
  7. Databases: 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, Database Recovery, Database Security & Authorization
  8. Algorithms: Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Dynamic Programming, Minimum Spanning Trees, Shortest Path Algos
  9. Theory of Computation: Introduction to TOC, Deterministic FA (DFA), Non-Deterministic FA, Regular Expressions, Grammar, Regular Language Properties, Moore & Mealy Machines, Pushdown Automata & CFG, Turing Machines, Complexity Theory
  10. 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
  11. Operating System: Introduction to OS, Process Management, CPU Scheduling, Process Synchronization, Threads & Process Creation, Deadlock, Memory Management, Virtual Memory, Disc Scheduling, File Management, Security, Windows OS, Distributed Systems, Unix OS, Linux OS
  12. Computer Networks: Introduction to CN, 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, Data Communication, Website, Browser & Services, Web Design & Development, Network Security, Cyber Threats and Malware Attacks, Cyber Safety and Security
  13. Full Length Mock Tests:
  14. Sectional Tests:
  15. Paper 2 - Full-length Mock Tests:
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

The video is a lecture on deriving inequalities for connected planar graphs using Euler's formula. The instructor, Sanchit Jain, explains how to derive two specific inequalities: $e \le 3v - 6$ and $r \le 2v - 4$. He starts by listing four points on a slide, explaining the relationship between edges and faces ($2e \ge 3r$) and the definition of region degree. He then proceeds to algebraically derive the first inequality by substituting Euler's formula into the edge-face inequality.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The instructor begins by presenting a slide titled 'Other formula derived from Euler's formula'. He lists four key points. Point 1 states that connected planar graphs with more than one edge obey $2e \ge 3r$. He explains the logic: each face has at least three face-edge incidences, and each edge contributes exactly two incidences. Point 2 defines the degree of a region as the number of edges covering it, noting the sum of degrees equals $2|E|$. He then focuses on Point 3, which claims that eliminating $r$ from $r = e - v + 2$ and $3r \le 2e$ results in $e \le 3v - 6$. He starts the derivation on the board, writing $r = e - v + 2$ as equation (1) and $r \le rac{2}{3}e$ as equation (2). He substitutes (1) into (2) to get $ rac{2}{3}e \ge e - v + 2$. Multiplying by 3 gives $2e \ge 3e - 3v + 6$. Rearranging terms leads to $-e \ge -3v + 6$, and finally $e \le 3v - 6$, which he boxes.

  2. 2:00 – 4:06 02:00-04:06

    Next, the instructor derives the formula for Point 4. He writes Euler's formula $r = e - v + 2$ again. He rearranges it to solve for $e$, obtaining $e = r + v - 2$. He then uses the inequality $3r \le 2e$, which can be rewritten as $e \ge rac{3}{2}r$. Substituting the expression for $e$ into this inequality gives $r + v - 2 \ge rac{3}{2}r$. To clear the fraction, he multiplies the entire inequality by 2, resulting in $2r + 2v - 4 \ge 3r$. He subtracts $2r$ from both sides to get $2v - 4 \ge r$, or $r \le 2v - 4$. He boxes this final result and points to the corresponding line on the slide, completing the derivation of the second inequality.

The lecture provides a rigorous algebraic derivation of two critical inequalities for connected planar graphs. Starting with the basic property that $2e \ge 3r$ due to face-edge incidences, the instructor combines this with Euler's formula ($r = e - v + 2$). By substituting and rearranging terms, he first eliminates the variable $r$ to prove that the number of edges is bounded by $e \le 3v - 6$. Subsequently, he eliminates $e$ to prove that the number of regions is bounded by $r \le 2v - 4$. These steps are clearly written on the whiteboard, showing the transition from Euler's formula to these specific constraints, which are essential for determining if a graph is planar.

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