[Graph Theory] Summary of Knowledge Points

This article is a summary of the knowledge points of graph theory.

Graph definition and terminology

[ Figure definition and terms ]

Storage structure

Commonly used graph storage structures are connection matrix and adjacency table. For details, see 【Chain Forward Star】.

A graph with relatively few edges is called a sparse graph , and an adjacency list is usually used . The adjacency matrix of a sparse graph is a sparse matrix, and there are special techniques to process it.

Those with more edges are called dense graphs . The adjacency matrix is ​​usually used.

Another method is the incidence matrix . The rows of the matrix represent vertices and the columns represent edges. If the edge is associated with a point, the element is 1.

Graph traversal

[ Picture traversal ]

Graph isomorphism

When two simple graphs are isomorphic, the vertices of the two graphs have a one-to-one correspondence (edge ​​correspondence) that maintains the adjacent relationship. That is to say, the graph structure is the same when the vertex identifier is ignored.

It is difficult to judge the graph isomorphism, but it is easier to judge the different structure of the graph. The properties maintained in the isomorphism of graphs are called graph invariants , and graph invariants can be used to judge the different structure of graphs. The same degree of corresponding vertices is a commonly used graph invariant.

You can define the mapping function from the graph to the graph vertex and write the adjacency matrix of the two graphs. If the matrices are equal, it means that the edges are preserved and the two graphs are isomorphic. But if the adjacency matrices are not equal, it cannot explain the different structure of the graph, because other mapping methods may make the adjacency matrices equal.

Connectivity

[ Connectivity of the graph ].

Euler Path and Hamilton Path

[ Euler Pathway and Hamilton Pathway ]

Directed Acyclic Graph and Its Application

[ Directed acyclic graph and its application ]

Shortest path

[ Shortest path algorithm ]

Floor plan

If a graph can be drawn in a plane without any intersection of edges, then the graph is a plane graph.

Euler formula

Let $r$ is the number of faces represented by the plane of the graph, then$r=e-v+2$。

inference

1. For connected plane graphs, if $v\ge 3$则 则$e\le 3v-6$.

2. There is no vertex with degree more than 5 in the simple graph of connected plane.

The above two inferences can be used to prove that the graph is a non-planar graph .

If the simple plane graph has no loop of length 3, then $e\le 2v-4$。

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