离散数学——Graph

目录

• Path and Degree
• Special graphs
• Bipartite graph
• Matching
• Operations on graph
• Representation of graph
• Isomorphism
• Connected Components
• Cut vertices and cut edges
• Counting paths
• special circuits and special path
  • Euler circuits and Euler path
  • Hamilton circuits and Hamilton path
• Single source shortest path
• Planar Graph
• Region
• Graph coloring
• Tree
  • definition
  • Theorem
  • Minimum Spanning Tree

Path and Degree

In a relation on set A, if there is a path between two nodes, then the path must have length \(l <= n-1\)

Handshaking theorem

The sum of all vertices' degree is an even number, therefore, the number of vertices with odd degree must be even

we use \(deg^-\) to represent in-degree, and \(deg^+\) to represent out-degree

Special graphs

We have the following special graphs

1. Complete graph \(K_n\): there is exactly one edge between every two vertices
2. Cycle \(C_n\): S = \(\{(v_1, v_2), (v_2, v_3), ... , (v_n, v_1)\}\)
3. Wheel graph \(W_n\): put one more vertex in the center of a cycle. Caution: A \(W_n\) contains \(n+1\) vertices
4. hypercube \(Q_n\): has \(2^n\) vertices, and all the vertices together they represent all bit strings of length \(n\), and every vertex has a different bit string. If two vertices' bit strings only differ in 1 bit, then they are adjacent in \(Q_n\), otherwise no

Bipartite graph

definition of bipartite graph

we can use only two color to paint bipartite graph

definition of complete bipartite graph

Matching

definition

definition of maximum matching and complete matching

We use Hall's Marriage Theorem to confirm a complete matching

Operations on graph

Representation of graph

Isomorphism

We can tell if two graphs are not isomorphic easily by finding special vertices

Existed cycles and path length won't vary in isomorphic graphs

Connected Components

definition

definition of strongly connected components and weakly connected components

Cut vertices and cut edges

Counting paths

we can use multiplication of adjacency matrix to find path

special circuits and special path

Euler circuits and Euler path

definition

Test of Euler circuit and Euler path

Euler circuit: every vertex must have even degree

Euler path: exactly two vertices have odd degree

Hamilton circuits and Hamilton path

definition

condition of Hamilton circuits

Single source shortest path

Dijkstra's Algorithm

Planar Graph

definition

test of planar graph

Region

degree of a region

Graph coloring

definition

Four Color Theorem

Tree

definition

definition of tree

definition of rooted tree

definition of level and height

Theorem

Minimum Spanning Tree

Prim's Algorithm and Kruscal's Algorithm