Data structure - basic concepts of graphs

1. Graph G consists of vertex set V and edge set E composed of G=(V,E), where V(G) represents the finite non-empty set of vertices in graph G; E (G) represents the set of relationships (edges) between vertices in graph G.

2. If $V=\left \{ v_{1},v_{2},...,v_{n} \right \}$ , then use |V| to represent the number of vertices in the graph G, also called the order of the graph G;

3. $E=\left \{ (u,v)| u\in V,v\in V \right \}$ , use |E| to represent the number of edges in graph G;

4. The linear list can be an empty list, and the tree can be an empty tree, but the graph cannot be an empty graph, that is, V must be a non-empty set, but E can be an empty set, as follows:

2. Undirected graph and directed graph

1. Undirected graph

If E is a finite set of undirected edges (edges for short), then the graph G is an undirected graph. An edge is an unordered pair of vertices, denoted as (v,w) or (w,v), because(v,w)=(w,v), where v and w are vertices. It can be said that vertex w and vertex v are adjacent points to each other, and edge (v, w) is attached to vertices w and v, or edge (v, w) is associated with vertices v and w. As shown below:

$G_{2}=(V_{2},E_{2})$

$V_{2}=\left \{ A,B,C,D,E \right \}$

$E_{2}=\left \{ (A,B),(B,D),(B,E),(C,D),(C,E),(D,E) \right \}$

2. Directed graph

If E is a finite set of directed edges (also called arcs), then the graph G is a directed graph. An arc is an ordered pair of vertices, denoted as <v,w>, where v and w are vertices, v is called the tail of the arc, and w is called the head of the arc, <v,w> is called the arc from vertex v to vertex w. It is also called v adjoining w, or w adjoining v. $<v,w>\neq <w,v>$ , as shown below:

$G_{1}=(V_{1},E_{1})$

$V_{1}=\left \{ A,B,C,D,E \right \}$

$E_{1}=\left \{ <A,B>,<A,C>,<A,D>,<A,E>,<B,A>,<B,C>,<B,E>,<C,D> \right \}$

3. Simple graphs and multiple graphs

1. Simple diagram:

(1) There are no duplicate edges;

(2) There is no edge from the vertex to itself;

(3)Main research department;

2.Multiple pictures:

(1) There is more than one edge between two nodes in graph G, and the vertices are allowed to be associated with themselves through the same edge, then G is a multigraph;

4. Degree, in-degree, out-degree of vertex

1. Degree of undirected graph

(1) Forundirected graph, the degree of vertex v refers to the number of edges attached to the vertex, denoted by isTD(v);

(2) In an undirected graph with n vertices and e edges, the sum of the degrees of all vertices is equal to twice the number of edges, that is:

$\sum_{i=1}^{n}TD(v_{i})=2e$

2. Degree, in-degree and out-degree of directed graph

（1）对于directional image对言，

In-degree is the number of directed edges ending with vertex v, recorded asID(v);

Out-degree is the number of directed edges starting from vertex v, recorded asOD(v);

(2) The degree of vertex v is equal to the sum of its in-degree and out-degree, that is,TD(v)=ID(v)+OD(v) ;

(3) In a directed graph with n vertices and e edges, the sum of in-degrees and out-degrees of all vertices are equal and equal to the number of edges, that is:

$\sum_{i=1}^{n}ID(v_{i})=\sum_{i=1}^{n}OD(v_{i})=e$

5. Description of the relationship between vertices

1.Path

A path from one vertex to another vertex refers to a sequence of vertices. There is no limit to the path direction of an undirected graph, but the path direction of a directed graph can only be consistent with the direction of the arc;

2.Circuit

The path in which the first vertex and the last vertex are the same is called a loop or loop;

3. Simple path

In the path sequence, a path whose vertices do not appear repeatedly is called a simple path;

4. Simple loop

A loop in which the vertices do not appear repeatedly except for the first vertex and the last vertex are called simple loops;

5. Path length

The number of edges in the path;

6. Distance from point to point

If the shortest path from vertex u to vertex v exists, the length of this path is called the distance from u to v; if there is no path from u to v at all, then the distance is recorded as infinity (∞);

7. Connected

a>connected;

(2)Directed graph, if there are paths from vertex v to vertex w and from vertex w to vertex v , then the two vertices are said to be strongly connected;

(3) If any two vertices in undirected graphG are connected, then the graph G is called Connected graph, otherwise it is calledDisconnected graph;

(4) If any pair of vertices in the directed graph is strongly connected, then the graph is called Strongly connected graph;

(5) For an undirected graph G with n vertices, if G is a connected graph, there will be at least n-1 edges; if G is a non-connected graph, there may be at most $C_{n-1}^{2}\textrm{}$ edge;

(6) For a directed graph G with n vertices, if G is a strongly connected graph, there will be at least n edges (forming a cycle);

6. Sub-picture

1.Definition

(1) There are two graphs G=(V,E) and G'=(V',E'), if V' is a subset of V, and E' is a subset of E, then G' is said to be a subgraph of G;To put it bluntly, it is to extract a part from the original picture, but this part must be called the upper picture. ;

(2) If there is a subgraph G' that satisfies V(G')=V(G), it is called a generated subgraph of G; To put it bluntly, all the vertices are picked out, and the edges are arbitrary;

2.Illustration

7. Connected components

1. Connected components

(1)The maximal connected subgraph in the undirected graph is called the connected component< /span>;

(2)The subgraph must be connected and contain as many vertices and edges as possible;

(3) Illustration:

2. Strongly connected components

(1)The maximal strongly connected subgraph in the directed graph is called the directed graphStrongly connected component;

(2)The subgraph must be strongly connected while retaining as many edges as possible;

(3) Illustration:

8. Spanning tree

1. Spanning tree

(1)The spanning tree of a connected graph is a minimal connected subgraph that contains all the vertices in the graph;

(2) There should be as few edges as possible, but they should be connected;

(3) If the number of vertices in the graph is n, its spanning tree contains n-1 edges. For a spanning tree, if one of its edges is cut off, it will become a non-connected graph, and if an edge is added, it will form a cycle;

(4)Illustration:

2. Generate a forest

(1) Innon-connected graph, the spanning tree of connected components constitutes the spanning forest of the non-connected graph;

(2) Illustration:

9. The rights of edges and the pictures/nets with rights

1. The right of the edge

In a graph, each edge can be marked with a value with a certain meaning, which is called the weight of the edge;

2. Authorized pictures/nets

A graph with weighted edges is called a weighted graph or network;

3. Weighted path length

When the graph is a weighted graph, the sum of the weights of all edges on a path is called the weighted path length of the path;

4.Illustration

10. Several special forms of pictures

1. Undirected complete graph

(1) Definition: There is an edge between any two vertices;

（2）边数应为 $C_{n}^{2}$ ；

(3) If the number of vertices of the undirected graph |V|=n, then $|E|\in [0,C_{n}^{2}]=[0,\frac{n(n-1)}{2}]$ ;

(4)Illustration:

2. Directed complete graph

(1) Definition: There are two arcs in opposite directions between any two vertices;

（2）边数应为 $2C_{n}^{2}$ ；

(3) If the number of vertices of the directed graph |V|=n, then $|E|\in [0,2C_{n}^{2}]=[0,n(n-1)]$ ;

(4)Illustration:

3. Sparse graphs and dense graphs

(1) Sparse graph: a graph with a small number of edges;

(2) Dense graph: a graph with many edges;

(3) There is no absolute limit. Generally speaking, when |E|<|V|log|V|, G can be regarded as a sparse graph;

4.Tree

(1) An undirected graph that does not exist and is connected;

(2) A tree with n vertices, must have n-1 edges;

(3) A graph with n vertices,If |E|>n-1, there must be a cycle;

(4) Directed tree: a directed graph in which the in-degree of one vertex is 0 and the in-degree of the other vertices is 1;

(5)Illustration: