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Introduction to graph theory and basic algorithms

Mikhail Tuzhilin, Dong Zhang

TL;DR

This book collects the lectures about graph theory and its applications which were given to students of mathematical departments of Moscow State University and Peking University to give students thorough knowledge about graphs to understand modern scientific fields more deeply.

Abstract

This book collects the lectures about graph theory and its applications which were given to students of mathematical departments of Moscow State University and Peking University. Graph theory is a very wide field with a lot of applications in almost every scientific area: in many branches of mathematics, computer science, physics, chemistry, biology and also in psychology, arts, philosophy and many others. Nowadays, graph theory becomes especially more important because of the rapid development of molecular biology, neural networks and AI fields. One of the aims of writing this book was to give students thorough knowledge about graphs to understand modern scientific fields more deeply. Here we tried to give classical and modern theorems and algorithms in more understandable and simple way. We spent many time to rewrite them and close the gaps in several simplest well-known proofs to provide more precise and accurate material for students.

Introduction to graph theory and basic algorithms

TL;DR

This book collects the lectures about graph theory and its applications which were given to students of mathematical departments of Moscow State University and Peking University to give students thorough knowledge about graphs to understand modern scientific fields more deeply.

Abstract

This book collects the lectures about graph theory and its applications which were given to students of mathematical departments of Moscow State University and Peking University. Graph theory is a very wide field with a lot of applications in almost every scientific area: in many branches of mathematics, computer science, physics, chemistry, biology and also in psychology, arts, philosophy and many others. Nowadays, graph theory becomes especially more important because of the rapid development of molecular biology, neural networks and AI fields. One of the aims of writing this book was to give students thorough knowledge about graphs to understand modern scientific fields more deeply. Here we tried to give classical and modern theorems and algorithms in more understandable and simple way. We spent many time to rewrite them and close the gaps in several simplest well-known proofs to provide more precise and accurate material for students.
Paper Structure (44 sections, 107 theorems, 161 equations, 35 figures)

This paper contains 44 sections, 107 theorems, 161 equations, 35 figures.

Table of Contents

  1. Basic definitions
  2. Two matrices corresponded to a graph
  3. Adjacency matrix
  4. Incidence matrix
  5. Connectivity and trees
  6. Spanning trees and algorithms
  7. Spanning trees of unweighted graphs
  8. Minimal spanning trees of weighted graphs
  9. Laplacian matrix and Kirchhoff's theorem
  10. Bipartite graph
  11. Shortest path problems
  12. Planar graph and polyhedrons
  13. Planar graph
  14. Polyhedrons
  15. $K$-connectivity
  16. ...and 29 more sections

Key Result

Lemma 1

(Handshaking lemma).

Figures (35)

  • Figure 1: Cube graph representation.
  • Figure 2: Example of isomorphism.
  • Figure 3: Example of non-isomorphic graphs.
  • Figure 4: Are these graphs isomorphic or not?
  • Figure 5: Graphs of hypercubes.
  • ...and 30 more figures

Theorems & Definitions (198)

  • Definition 1.1
  • Definition 1.2
  • Example 1.1
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Example 1.2
  • Definition 1.6
  • Definition 1.7
  • Definition 1.8
  • ...and 188 more