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BCA-411E · Discrete Mathematics
Unit 5

Graph Theory, Trees & Network Algorithms

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Unit 5 Learning Hub

Graph Theory, Trees & Network Algorithms

Official MJPRU BCA Unit 5: Graph Theory, Trees & Network Algorithms for BCA-411E (Discrete Mathematics).

Syllabus Topics:Graph Fundamentals: Definition G = (V, E), Vertices, Edges, Directed Graphs (Digraphs) vs Undirected GraphsGraph Terminology: Adjacent Vertices, Incident Edges, Degree of a Vertex (deg(v)), In-degree and Out-degreeHandshaking Theorem (Euler): Sum of Degrees of all Vertices = 2 * |E|, Proof and Corollaries (Number of Odd Vertices is Even)Types of Graphs: Simple Graphs, Multigraphs, Pseudographs (Self-Loops and Parallel Edges)Special Families of Simple Graphs: Complete Graphs (K_n), Cycle Graphs (C_n), Wheel Graphs (W_n), Regular Graphs (k-regular)Bipartite Graphs: Definition, Characterization (Graph is Bipartite iff it has No Odd Cycles), Complete Bipartite Graphs (K_{m,n})Subgraphs, Isomorphic Graphs, and Homeomorphic Graphs: Invariants of Graph IsomorphismWalks, Trails, Paths, and Cycles: Connected Graphs, Connected Components, Cut-Points (Articulation Points) and BridgesEulerian Graphs: Eulerian Trail, Eulerian Circuit, Euler's Theorem (Connected Graph is Eulerian iff Every Vertex has Even Degree)Hamiltonian Graphs: Hamiltonian Path, Hamiltonian Cycle, Dirac's Theorem and Ore's Theorem CriteriaPlanar Graphs: Euler's Planar Formula V - E + F = 2, Non-Planar Graphs, Kuratowski's Theorem (K_5 and K_{3,3})Graph Coloring: Vertex Coloring, Chromatic Number chi(G), The Famous Four Color TheoremTree Fundamentals: Definition as Connected Acyclic Graph, Equivalent Characterizations of Trees (n vertices, n-1 edges)Spanning Trees: Definition, Existence in Connected Graphs, Fundamental Cycles and Fundamental CutsetsMinimum Spanning Tree (MST) Algorithms: Kruskal's Greedy Algorithm (Disjoint Set Union) and Prim's AlgorithmMatrix Representations of Graphs: Adjacency Matrix (A), Properties of Adjacency Matrix Powers, Incidence Matrix (M)
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Introduction to Computers & The IPO Cycle

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Topic 1 · The foundation of modern computing

Introduction to Computers & The IPO Cycle

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💡 The Big Picture:A computer is an electronic system that takes raw data as Input, processes it under programmed logic, and produces meaningful Information as Output.
🍪 Everyday Analogy:Think of baking cookies: Flour & Sugar (Input) ➔ The Oven (Processing) ➔ Warm Delicious Cookie (Output)! 🍪
Key Exam Takeaways
  • ·Data = Raw, unprocessed facts without context (e.g. 42, 'apple').
  • ·Information = Structured data with meaning (e.g. 'Temperature: 42°C').
  • ·Hardware is what you can touch; Software is the instructions running inside.
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Which of the following represents unprocessed raw facts?

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