有向图，无向图的邻接矩阵和邻接表模板

图

- 图的定义
- - 有向图
  - - 概念
    - 模板
    - - 邻接矩阵
      - 邻接表
  - 无向图
  - - 概念
    - 模板
    - - 邻接矩阵
      - 邻接表
  - 简单图
  - 完全图

图的定义

图 $G$ 由顶点集 $V$ 和边集 $E$ 组成，记为 $G = (V, E)$ ，其中 $V (G)$ 表示图 $G$ 中顶点的有限非空集； $E (G)$ 表示图 $G$ 中顶点之间的关系（边）集合。

有向图

概念

若 $E$ 是有向边（也称弧）的有限集合时，则图 $G$ 为有向图。弧是顶点的有序对，记为 $< v, w >$ ，其中 $v, w$ 是顶点， $v$ 称为弧尾， $w$ 称为弧头， $< v, w >$ 称为从 $v$ 到 $w$ 的弧，也称 $v$ 邻接到 $w$ 。

模板

邻接矩阵

/* 
有向图邻接矩阵
*/

#include <iostream>
#include <cstring>
#include <algorithm>
#include <vector>
using namespace std;

#define MaxVertexNum 100
typedef int weightType;  // 权重数据类型
typedef char vertexType; // 顶点数据类型

struct Graph
{
    
    
    int vertexnum;
    int edgenum;
    vertexType vertexList[MaxVertexNum];
    weightType edgeList[MaxVertexNum][MaxVertexNum];
};

void BuildGraph(Graph *G)
{
    
    
    int start, end;
    cout << "Please enter the number of vertices and edges" << endl;
    cin >> G->vertexnum >> G->edgenum;
    // 图的权重初始化
    for (int i = 0; i < G->vertexnum; i++)
    {
    
    
        for (int j = 0; j < G->vertexnum; j++)
        {
    
    
            G->edgeList[i][j] = 0;
        }
    }
    // 图的顶点数据
    for (int i = 0; i < G->vertexnum; i++)
    {
    
    
        cout << "Please enter the data of vertex" << i + 1 << endl;
        cin >> G->vertexList[i];
    }
    // 输入权重信息
    for (int i = 0; i < G->edgenum; i++)
    {
    
    
        cout << "Please enter the Start number, end number, weight" << endl;
        cin >> start >> end;
        cin >> G->edgeList[start - 1][end - 1];
    }
    cout << endl;
}

void Print_Adjacency_Matrix(Graph G)
{
    
    
    for (int i = 0; i < G.vertexnum; i++)
    {
    
    
        for (int j = 0; j < G.vertexnum; j++)
        {
    
    
            cout << G.edgeList[i][j] << '\t';
        }
        cout << endl;
    }
}

int main()
{
    
    
    Graph G;
    BuildGraph(&G);
    Print_Adjacency_Matrix(G);
    cout << endl;
    system("pause");
    return 0;
}

邻接表

/* 
有向图邻接表
*/

#include <iostream>
#include <cstring>
#include <algorithm>
#include <vector>
using namespace std;

#define MaxVertexNum 100
typedef int weightType;  // 权重数据类型
typedef char vertexType; // 顶点数据类型

struct EdgeNode
{
    
    
    int adjvex;
    weightType weight;
    EdgeNode *next;
};

struct VertexNode
{
    
    
    vertexType data;
    EdgeNode *firstedge;
};

struct Graph
{
    
    
    int vertexnum;
    int edgenum;
    VertexNode vertexList[MaxVertexNum];
    VertexNode inverseVertexList[MaxVertexNum];
};

void BuildGraph(Graph *G)
{
    
    
    int start, end, weight;
    EdgeNode *newnode;
    cout << "Please enter the number of vertices and edges" << endl;
    cin >> G->vertexnum >> G->edgenum;
    // 图的顶点数据
    for (int i = 0; i < G->vertexnum; i++)
    {
    
    
        cout << "Please enter the data of vertex" << i << endl;
        cin >> G->vertexList[i].data;
        G->vertexList[i].firstedge = NULL;
        G->inverseVertexList[i].data = G->vertexList[i].data;
        G->inverseVertexList[i].firstedge = NULL;
    }
    // 输入权重信息
    for (int i = 0; i < G->edgenum; i++)
    {
    
    
        cout << "Please enter the Start number, end number, weight" << endl;
        cin >> start >> end >> weight;
        //start-->end
        newnode = new EdgeNode;
        newnode->adjvex = end;
        newnode->weight = weight;
        newnode->next = G->vertexList[start].firstedge;
        G->vertexList[start].firstedge = newnode;
        // 逆邻接表
        newnode = new EdgeNode;
        newnode->adjvex = start;
        newnode->weight = weight;
        newnode->next = G->inverseVertexList[end].firstedge;
        G->inverseVertexList[end].firstedge = newnode;
    }
}

void Print_Adjacency_Matrix(Graph G)
{
    
    
    for (int i = 0; i < G.vertexnum; i++)
    {
    
    
        cout << G.vertexList[i].data << '\t';
        EdgeNode *p = G.vertexList[i].firstedge;
        while (p)
        {
    
    
            printf("adjvex:%d weight:%d  ", p->adjvex, p->weight);
            p = p->next;
        }
        cout << endl;
    }
    cout << endl;
    cout << "inverse adjacency list" << endl;
    for (int i = 0; i < G.vertexnum; i++)
    {
    
    
        cout << G.inverseVertexList[i].data << '\t';
        EdgeNode *p = G.inverseVertexList[i].firstedge;
        while (p)
        {
    
    
            printf("adjvex:%d weight:%d  ", p->adjvex, p->weight);
            p = p->next;
        }
        cout << endl;
    }
}

int main()
{
    
    
    Graph G;
    BuildGraph(&G);
    Print_Adjacency_Matrix(G);
    cout << endl;
    system("pause");
    return 0;
}

无向图

概念

若 $E$ 是无向边的有限集合时，则图 $G$ 为有向图。边是顶点的无序对，记为 $(v, w)$ 或 $(w, v)$ 。可以说 $w$ 和 $v$ $互为$ 邻接点。边 $(V, W)$ 依附于 $w$ $和 v$ ，或称边 $(v, w)$ 和 $v, w$ 相关联。

模板

邻接矩阵

/* 
无向图邻接矩阵
*/

#include <iostream>
#include <cstring>
#include <algorithm>
#include <vector>
using namespace std;

#define MaxVertexNum 100
typedef int weightType;  // 权重数据类型
typedef char vertexType; // 顶点数据类型

struct Graph
{
    
    
    int vertexnum;
    int edgenum;
    vertexType vertexList[MaxVertexNum];
    weightType edgeList[MaxVertexNum][MaxVertexNum];
};

void BuildGraph(Graph *G)
{
    
    
    int start, end;
    cout << "Please enter the number of vertices and edges" << endl;
    cin >> G->vertexnum >> G->edgenum;
    // 图的权重初始化
    for (int i = 0; i < G->vertexnum; i++)
    {
    
    
        for (int j = 0; j < G->vertexnum; j++)
        {
    
    
            G->edgeList[i][j] = 0;
        }
    }
    // 图的顶点数据
    for (int i = 0; i < G->vertexnum; i++)
    {
    
    
        cout << "Please enter the data of vertex" << i + 1 << endl;
        cin >> G->vertexList[i];
    }
    // 输入权重信息
    for (int i = 0; i < G->edgenum; i++)
    {
    
    
        cout << "Please enter the Start number, end number, weight" << endl;
        cin >> start >> end;
        cin >> G->edgeList[start - 1][end - 1];
        G->edgeList[end - 1][start - 1] = G->edgeList[start - 1][end - 1];
    }
    cout << endl;
}

void Print_Adjacency_Matrix(Graph G)
{
    
    
    cout << '\t';
    for (int i = 0; i < G.vertexnum; i++)
    {
    
    
        cout << G.vertexList[i] << '\t';
    }
    cout << endl;
    for (int i = 0; i < G.vertexnum; i++)
    {
    
    
        cout << G.vertexList[i] << '\t';
        for (int j = 0; j < G.vertexnum; j++)
        {
    
    
            cout << G.edgeList[i][j] << '\t';
        }
        cout << endl;
    }
}

int main()
{
    
    
    Graph G;
    BuildGraph(&G);
    Print_Adjacency_Matrix(G);
    cout << endl;
    system("pause");
    return 0;
}

邻接表

/* 
无向图邻接表
*/

#include <iostream>
#include <cstring>
#include <algorithm>
#include <vector>
using namespace std;

#define MaxVertexNum 100
typedef int weightType;  // 权重数据类型
typedef char vertexType; // 顶点数据类型

struct EdgeNode
{
    
    
    int adjvex;
    weightType weight;
    EdgeNode *next;
};

struct VertexNode
{
    
    
    vertexType data;
    EdgeNode *firstedge;
};

struct Graph
{
    
    
    int vertexnum;
    int edgenum;
    VertexNode vertexList[MaxVertexNum];
};

void BuildGraph(Graph *G)
{
    
    
    int start, end, weight;
    EdgeNode *newnode;
    cout << "Please enter the number of vertices and edges" << endl;
    cin >> G->vertexnum >> G->edgenum;
    // 图的顶点数据
    for (int i = 0; i < G->vertexnum; i++)
    {
    
    
        cout << "Please enter the data of vertex" << i << endl;
        cin >> G->vertexList[i].data;
        G->vertexList[i].firstedge = NULL;
    }
    // 输入权重信息
    for (int i = 0; i < G->edgenum; i++)
    {
    
    
        cout << "Please enter the Start number, end number, weight" << endl;
        cin >> start >> end >> weight;
        //start-->end
        newnode = new EdgeNode;
        newnode->adjvex = end;
        newnode->weight = weight;
        newnode->next = G->vertexList[start].firstedge;
        G->vertexList[start].firstedge = newnode;
        // end-->start
        newnode = new EdgeNode;
        newnode->adjvex = start;
        newnode->weight = weight;
        newnode->next = G->vertexList[end].firstedge;
        G->vertexList[end].firstedge = newnode;
    }
}

void Print_Adjacency_Matrix(Graph G)
{
    
    
    for (int i = 0; i < G.vertexnum; i++)
    {
    
    
        cout << G.vertexList[i].data << '\t';
        EdgeNode *p = G.vertexList[i].firstedge;
        while (p)
        {
    
    
            printf("adjvex:%d weight:%d\n", p->adjvex, p->weight);
            p = p->next;
        }
        cout << endl;
    }
}

int main()
{
    
    
    Graph G;
    BuildGraph(&G);
    Print_Adjacency_Matrix(G);
    cout << endl;
    system("pause");
    return 0;
}

简单图

不存在重复边
不存在顶点到自身的边

完全图

对于无向图

${E}\vert$ 的取值范围 $0$ 到 $n (n - 1) / 2$ ，有 $n (n - 1) / 2$ 条边的无向图称为完全图

对于有向图

${E}\vert$ 的取值范围 $0$ 到 $n (n - 1)$ ，有 $n (n - 1)$ 条弧的有向图称为有向完全图