Graph Theory Fundamentals in Theory of Computation

Introduction

Section 1: Basic Concepts

1.1 Definition of a Graph

• Example: Consider a graph \(G\) with vertices \(V = \{A, B, C\}\) and edges \(E = \{(A, B), (B, C), (C, A)\}\). This forms a triangle where each vertex is connected to the other two.

1.2 Types of Graphs

1.2.1 Directed vs. Undirected Graphs

• Undirected Graph: A graph where the edges have no direction. The edge \((A, B)\) is the same as \((B, A)\).
  • Example: A social network where friendship is mutual.
• Directed Graph (Digraph): A graph where the edges have a direction. The edge \((A, B)\) is not the same as \((B, A)\).
  • Example: A food web where arrows indicate predator-prey relationships.

1.2.2 Weighted Graphs

• Weighted Graph: A graph where each edge has an associated weight or cost.
  • Example: A road network where weights represent distances or travel times.

1.2.3 Connected vs. Disconnected Graphs

• Connected Graph: A graph where there is a path between every pair of vertices.
  • Example: A computer network where every computer can communicate with every other computer.
• Disconnected Graph: A graph where there are at least two vertices with no path between them.
  • Example: A group of islands where some islands are not reachable by bridges or ferries.

Section 2: Graph Representations

2.1 Adjacency Matrix

• Example:
  • For a graph with vertices \(V = \{A, B, C\}\) and edges \(E = \{(A, B), (B, C), (C, A)\}\), the adjacency matrix would be:

    A B C
    A 0 1 1
    B 0 0 1
    C 1 0 0

  2.2 Adjacency List

  An adjacency list is an array of lists, where each list contains the neighbors of a vertex.
  • Example:

    • A: B, C
      B: C
      C: A

  Section 3: Applications in TOC

  3.1 State Transition Diagrams in Automata

  In automata theory, state transition diagrams are used to represent the behavior of finite automata. These diagrams are essentially directed graphs where vertices represent states and edges represent transitions based on input symbols.
  • • • Example: A finite automaton that accepts strings ending with ‘ab’ can be represented as:

      • digraph FA {
            node [shape=circle];
            A [label="A (Start)"];
            B [label="B"];
            C [label="C (Accept)"];
            A -> B [label="a"];
            B -> C [label="b"];
            C -> C [label="a,b"];
        }

  3.2 Algorithm Visualization

  Graph theory is also used to visualize the flow of algorithms, such as graph traversal algorithms (BFS, DFS) and shortest path algorithms (Dijkstra’s algorithm).

  Note to the Reader

  This post, “TOC Unit 1.1.3: Graph Theory Fundamentals for Computation,” provides a comprehensive overview of the essential concepts in graph theory and their applications in the Theory of Computation. Understanding these concepts is crucial for modeling and analyzing computational processes. In the next post, we will explore the concepts of strings, alphabets, and operations in TOC.
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