add C code for graph (#583)

* Create chapter_graph

* Delete chapter_graph

* add C code for graph

* add C code for graph

* Create graph_adjacency_list.c

add C code for graph

* Update CMakeLists.txt

* Update format and output

* Update format and output

* Update format and output

* Update format and output

* Update format and output

* Update format and output

* Update format and output

* Update format and output

* Update format and output

* Update format and output

* Update format and output

* Update format and output

* Update format and output

* Update format and output

* Update format and output

* Update graph_adjacency_list.c

* Update CMakeLists.txt for c code of graph

* Update format of c code

* Update format of c code

* Update format of c code

* Update verticesList

Change the data structure of the storage list from a linked list to a linear table

* Update graph_adjacency_list.c

* Update graph_adjacency_matrix.c

* Create graph_adjacency_list_test.c

* Create graph_bfs

* Update CMakeLists.txt

* Update graph_adjacency_list.c

codes/c/CMakeLists.txt

@@ -13,3 +13,4 @@ add_subdirectory(chapter_hashing)
 add_subdirectory(chapter_tree)
 add_subdirectory(chapter_searching)
 add_subdirectory(chapter_sorting)
+add_subdirectory(chapter_graph)

codes/c/chapter_graph/CMakeLists.txt

@@ -0,0 +1,4 @@
+add_executable(graph_adjacency_matrix graph_adjacency_matrix.c)
+add_executable(graph_adjacency_list graph_adjacency_list.c)
+add_executable(graph_adjacency_list_test graph_adjacency_list_test.c)
+add_executable(graph_bfs graph_bfs.c)

codes/c/chapter_graph/graph_adjacency_list.c

@@ -0,0 +1,301 @@
+/**
+ * File: graph_adjacency_list.c
+ * Created Time: 2023-07-07
+ * Author: NI-SW (947743645@qq.com)
+ */
+
+#include "../utils/common.h"
+
+typedef struct Vertex Vertex;
+typedef struct Node Node;
+typedef struct linkList linkList;
+
+void freeVertex(Vertex *);
+void freeLinklist(linkList *);
+linkList *newLinklist(Vertex *);
+
+/* 链表节点 */
+struct Node {
+    // 链表节点内包含顶点类和下一个节点地址
+    Vertex *val;
+    Node *next;
+};
+
+/* 链表节点构造函数 */
+Node *newNode() {
+    Node *n = (Node *)malloc(sizeof(Node));
+    n->next = 0;
+    n->val = 0;
+    return n;
+}
+
+/* 顶点节点类 */
+struct Vertex {
+    // 节点值
+    int val;
+    // 与其它节点相连接的边的链表
+    linkList *linked;
+    // 索引位，标记该顶点在顶点列表中的索引
+    unsigned int pos;
+};
+
+/* 顶点节点构造函数 */
+Vertex *newVertex(int val) {
+    Vertex *v = (Vertex *)malloc(sizeof(Vertex));
+    // 为新节点赋值并建立该节点的链表
+    v->val = val;
+    v->linked = newLinklist(v);
+    return v;
+}
+
+/* 顶点内存释放函数 */
+void freeVertex(Vertex *val) {
+    // 释放该顶点和该顶点的链表的内存
+    freeLinklist(val->linked);
+    free(val);
+}
+
+/* 链表 */
+struct linkList {
+    Node *head;
+    Node *tail;
+};
+
+/* 链表头插法 */
+void pushFront(linkList *l, Vertex *val) {
+    Node *temp = newNode();
+    temp->val = val;
+    temp->next = l->head->next;
+    l->head->next = temp;
+    if (l->tail == l->head) {
+        l->tail = temp;
+    }
+}
+
+/* 链表尾插法 */
+void pushBack(linkList *l, Vertex *val) {
+    Node *temp = newNode();
+    temp->val = val;
+    temp->next = 0;
+    l->tail->next = temp;
+    l->tail = temp;
+}
+
+/* 根据顶点地址与该顶点连接的删除边 */
+void removeLink(linkList *l, Vertex *val) {
+    Node *temp = l->head->next;
+    Node *front = l->head;
+    while (temp != 0) {
+        if (temp->val == val) {
+            front->next = temp->next;
+            if (l->tail == temp) {
+                l->tail = front;
+            }
+            free(temp);
+            return;
+        }
+        front = temp;
+        temp = temp->next;
+    }
+
+    if (temp->next == 0) {
+        printf("vertex not found!\n");
+    }
+}
+
+/* 根据索引查找链表中节点 */
+Node *findByindex(linkList *l, unsigned int index) {
+    unsigned int i = 0;
+    Node *temp = l->head->next;
+    while (temp != 0) {
+        if (i == index) {
+            return temp;
+        }
+        temp = temp->next;
+        i++;
+    }
+
+    if (temp->next == 0) {
+        printf("vertex not found!\n");
+        return 0;
+    }
+    return 0;
+}
+
+/* 根据顶点地址删除顶点 */
+void removeNode(linkList *l, Vertex *val) {
+    Node *temp = l->head->next;
+    Node *front = l->head;
+    while (temp != 0) {
+        if (temp->val == val) {
+            front->next = temp->next;
+            if (l->tail == temp) {
+                l->tail = front;
+            }
+            freeVertex(val);
+            free(temp);
+            return;
+        }
+        front = temp;
+        temp = temp->next;
+    }
+
+    if (temp->next == 0) {
+        printf("vertex not found!\n");
+    }
+}
+
+/* 释放链表内存 */
+void freeLinklist(linkList *l) {
+    Node *temp = l->head->next;
+    while (temp != 0) {
+        free(l->head);
+        l->head = temp;
+        temp = temp->next;
+    }
+    free(l->head);
+    l->head = 0;
+    free(l);
+}
+
+/* 链表构造函数 */
+linkList *newLinklist(Vertex *val) {
+    linkList *newLinklist = (linkList *)malloc(sizeof(linkList));
+
+    newLinklist->head = newNode();
+    newLinklist->head->val = val;
+    newLinklist->tail = newLinklist->head;
+    newLinklist->head->next = 0;
+
+    return newLinklist;
+}
+
+/* 基于邻接链表实现的无向图类结构 */
+struct graphAdjList {
+    // 顶点列表
+    Vertex **verticesList;
+    // 顶点数量
+    unsigned int size;
+    // 当前容量
+    unsigned int capacity;
+};
+
+typedef struct graphAdjList graphAdjList;
+
+/* 添加边 */
+void addEdge(graphAdjList *t, int i, int j) {
+    // 越界检查
+    if (i < 0 || j < 0 || i == j || i >= t->size || j >= t->size) {
+        printf("Out of range in %s:%d\n", __FILE__, __LINE__);
+        return;
+    }
+    // 查找待连接的节点
+    Vertex *v1 = t->verticesList[i];
+    Vertex *v2 = t->verticesList[j];
+
+    // 连接节点
+    pushBack(v1->linked, v2);
+    pushBack(v2->linked, v1);
+}
+
+/* 删除边 */
+void removeEdge(graphAdjList *t, int i, int j) {
+    // 越界检查
+    if (i < 0 || j < 0 || i == j || i >= t->size || j >= t->size) {
+        printf("Out of range in %s:%d\n", __FILE__, __LINE__);
+        return;
+    }
+
+    // 查找待删除边的相关节点
+    Vertex *v1 = t->verticesList[i];
+    Vertex *v2 = t->verticesList[j];
+
+    // 移除待删除边
+    removeLink(v1->linked, v2);
+    removeLink(v2->linked, v1);
+}
+
+/* 添加顶点 */
+void addVertex(graphAdjList *t, int val) {
+    // 若大小超过容量，则扩容
+    if (t->size >= t->capacity) {
+        Vertex **tempList = (Vertex **)malloc(sizeof(Vertex *) * 2 * t->capacity);
+        memcpy(tempList, t->verticesList, sizeof(Vertex *) * t->size);
+        free(t->verticesList);
+        // 指向新顶点表
+        t->verticesList = tempList;    
+        t->capacity = t->capacity * 2;
+    }
+    // 申请新顶点内存并将新顶点地址存入顶点列表
+    Vertex *newV = newVertex(val);
+    newV->pos = t->size;
+    newV->linked = newLinklist(newV);
+    t->verticesList[t->size] = newV;
+    t->size++;
+}
+
+/* 删除顶点 */
+void removeVertex(graphAdjList *t, unsigned int index) {
+    // 越界检查
+    if (index < 0 || index >= t->size) {
+        printf("Out of range in %s:%d\n", __FILE__, __LINE__);
+        exit(1);
+    }
+
+    // 查找待删节点
+    Vertex *v = t->verticesList[index];
+    // 若不存在该节点，则返回
+    if (v == 0) {
+        printf("index is:%d\n", index);
+        printf("Out of range in %s:%d\n", __FILE__, __LINE__);
+        return;
+    }
+
+    // 遍历待删除节点链表，将所有与待删除结点有关的边删除
+    Node *temp = v->linked->head->next;
+    while (temp != 0) {
+        removeLink(temp->val->linked, v);
+        temp = temp->next;
+    }
+
+    // 定点列表前移
+    for (int i = index; i < t->size - 1; i++) {
+        t->verticesList[i] = t->verticesList[i + 1];
+    }
+    t->verticesList[t->size - 1] = 0;
+    t->size--;
+
+    //释放被删除顶点的内存
+    freeVertex(v);
+}
+
+/* 打印顶点与邻接矩阵 */
+void printGraph(graphAdjList *t) {
+    printf("邻接表  =\n");
+    for (int i = 0; i < t->size; i++) {
+        Node *n = t->verticesList[i]->linked->head->next;
+        printf("%d: [", t->verticesList[i]->val);
+        while (n != 0) {
+            if (n->next != 0) {
+                printf("%d, ", n->val->val);
+            } else {
+                printf("%d", n->val->val);
+            }
+            n = n->next;
+        }
+        printf("]\n");
+    }
+}
+
+/* 构造函数 */
+graphAdjList *newGraphic(unsigned int verticesNumber) {
+    // 申请内存
+    graphAdjList *newGraph = (graphAdjList *)malloc(sizeof(graphAdjList));
+    // 建立顶点表并分配内存
+    newGraph->verticesList = (Vertex **)malloc(sizeof(Vertex *) * verticesNumber);
+    memset(newGraph->verticesList, 0, sizeof(Vertex *) * verticesNumber);
+    // 初始化大小和容量
+    newGraph->size = 0;
+    newGraph->capacity = verticesNumber;
+    return newGraph;
+}

codes/c/chapter_graph/graph_adjacency_list_test.c

@@ -0,0 +1,54 @@
+/**
+ * File: graph_adjacency_list_test.c
+ * Created Time: 2023-07-11
+ * Author: NI-SW (947743645@qq.com)
+ */
+
+#include "graph_adjacency_list.c"
+
+/* Driver Code */
+int main() {
+
+    /* 初始化无向图 */
+    graphAdjList *graph = newGraphic(5);
+    // 初始化顶点
+    addVertex(graph, 1);
+    addVertex(graph, 3);
+    addVertex(graph, 2);
+    addVertex(graph, 5);
+    addVertex(graph, 4);
+    // 初始化边
+    addEdge(graph, 0, 1);
+    addEdge(graph, 0, 3);
+    addEdge(graph, 1, 2);
+    addEdge(graph, 2, 3);
+    addEdge(graph, 2, 4);
+    addEdge(graph, 3, 4);
+    printf("\n初始化后，图为:\n");
+    printGraph(graph);
+
+    /* 添加边 */
+    // 顶点 1, 2 的索引分别为 0, 2
+    addEdge(graph, 0, 2);
+    printf("\n添加边 1-2 后图为\n");
+    printGraph(graph);
+
+    /* 删除边 */
+    // 顶点 1, 3 的索引分别为 0, 1
+    removeEdge(graph, 0, 1);
+    printf("\n删除边 1-3 后，图为\n");
+    printGraph(graph);
+
+    /* 添加顶点 */
+    addVertex(graph, 6);
+    printf("\n添加顶点 6 后，图为\n");
+    printGraph(graph);
+
+    /* 删除顶点 */
+    // 顶点 3 的索引为 1
+    removeVertex(graph, 1);
+    printf("\n删除顶点 3 后，图为\n");
+    printGraph(graph);
+
+    return 0;
+}

codes/c/chapter_graph/graph_adjacency_matrix.c

@@ -0,0 +1,238 @@
+/**
+ * File: graph_adjacency_matrix.c
+ * Created Time: 2023-07-06
+ * Author: NI-SW (947743645@qq.com)
+ */
+
+#include "../utils/common.h"
+
+/* 基于邻接矩阵实现的无向图类结构 */
+struct graphAdjMat {
+    int *vertices;
+    unsigned int **adjMat;
+    unsigned int size;
+    unsigned int capacity;
+};
+
+typedef struct graphAdjMat graphAdjMat;
+
+/* 添加边 */
+void addEdge(graphAdjMat *t, int i, int j) {
+    // 越界检查
+    if (i < 0 || j < 0 || i >= t->size || j >= t->size || i == j) {
+        printf("Out of range in %s:%d\n", __FILE__, __LINE__);
+        exit(1);
+    }
+    t->adjMat[i][j] = 1;
+    t->adjMat[j][i] = 1;
+}
+
+/* 删除边 */
+void removeEdge(graphAdjMat *t, int i, int j) {
+    // 越界检查
+    if (i < 0 || j < 0 || i >= t->size || j >= t->size || i == j) {
+        printf("Out of range in %s:%d\n", __FILE__, __LINE__);
+        exit(1);
+    }
+    t->adjMat[i][j] = 0;
+    t->adjMat[j][i] = 0;
+}
+
+/* 添加顶点 */
+void addVertex(graphAdjMat *t, int val) {
+    // 如果实际使用不大于预设空间，则直接初始化新空间
+    if (t->size < t->capacity) {
+        t->vertices[t->size] = val;
+
+        // 邻接矩新列阵置0
+        for (int i = 0; i < t->size; i++) {
+            t->adjMat[i][t->size] = 0;
+        }
+        memset(t->adjMat[t->size], 0, sizeof(unsigned int) * (t->size + 1));
+        t->size++;
+        return;
+    }
+
+    // 扩容，申请新的顶点数组
+    int *temp = (int *)malloc(sizeof(int) * (t->size * 2));
+    memcpy(temp, t->vertices, sizeof(int) * t->size);
+    temp[t->size] = val;
+
+    // 释放原数组
+    free(t->vertices);
+    t->vertices = temp;
+
+    // 扩容，申请新的二维数组
+    unsigned int **tempMat = (unsigned int **)malloc(sizeof(unsigned int *) * t->size * 2);
+    unsigned int *tempMatLine = (unsigned int *)malloc(sizeof(unsigned int) * (t->size * 2) * (t->size * 2));
+    memset(tempMatLine, 0, sizeof(unsigned int) * (t->size * 2) * (t->size * 2));
+    for (int k = 0; k < t->size * 2; k++) {
+        tempMat[k] = tempMatLine + k * (t->size * 2);
+    }
+
+    // 原数据复制到新数组
+    for (int i = 0; i < t->size; i++) {
+        memcpy(tempMat[i], t->adjMat[i], sizeof(unsigned int) * t->size);
+    }
+
+    // 新列置0
+    for (int i = 0; i < t->size; i++) {
+        tempMat[i][t->size] = 0;
+    }
+    memset(tempMat[t->size], 0, sizeof(unsigned int) * (t->size + 1));
+
+    // 释放原数组
+    free(t->adjMat[0]);
+    free(t->adjMat);
+
+    // 扩容后，指向新地址
+    t->adjMat = tempMat;
+    t->capacity = t->size * 2;
+    t->size++;
+}
+
+/* 删除顶点 */
+void removeVertex(graphAdjMat *t, unsigned int index) {
+    // 越界检查
+    if (index < 0 || index >= t->size) {
+        printf("Out of range in %s:%d\n", __FILE__, __LINE__);
+        exit(1);
+    }
+
+    // 清除删除的顶点，并将其后所有顶点前移
+    for (int i = index; i < t->size - 1; i++) {
+        t->vertices[i] = t->vertices[i + 1];
+    }
+
+    // 将被前移的最后一个顶点置0
+    t->vertices[t->size - 1] = 0;
+
+    // 清除邻接矩阵中删除的列
+    for (int i = 0; i < t->size - 1; i++) {
+        if (i < index) {
+            // 被删除列后的所有列前移
+            for (int j = index; j < t->size - 1; j++) {
+                t->adjMat[i][j] = t->adjMat[i][j + 1];
+            }
+        } else {
+            // 被删除行的下方所有行上移
+            memcpy(t->adjMat[i], t->adjMat[i + 1], sizeof(unsigned int) * t->size);
+            // 被删除列后的所有列前移
+            for (int j = index; j < t->size; j++) {
+                t->adjMat[i][j] = t->adjMat[i][j + 1];
+            }
+        }
+    }
+    t->size--;
+}
+
+/* 打印顶点与邻接矩阵 */
+void printGraph(graphAdjMat *t) {
+    if (t->size == 0) {
+        printf("graph is empty\n");
+        return;
+    }
+    printf("顶点列表 = [");
+    for (int i = 0; i < t->size; i++) {
+        if (i != t->size - 1) {
+            printf("%d, ", t->vertices[i]);
+        } else {
+            printf("%d", t->vertices[i]);
+        }
+    }
+    printf("]\n");
+    printf("邻接矩阵 =\n[\n");
+    for (int i = 0; i < t->size; i++) {
+        printf("  [");
+        for (int j = 0; j < t->size; j++) {
+            if (j != t->size - 1) {
+                printf("%u, ", t->adjMat[i][j]);
+            } else {
+                printf("%u", t->adjMat[i][j]);
+            }
+        }
+        printf("],\n");
+    }
+    printf("]\n");
+}
+
+/* 构造函数 */
+graphAdjMat *newGraphic(unsigned int numberVertices, int *vertices, unsigned int **adjMat) {
+    // 函数指针
+    graphAdjMat *newGraph = (graphAdjMat *)malloc(sizeof(graphAdjMat));
+
+    // 申请内存
+    newGraph->vertices = (int *)malloc(sizeof(int) * numberVertices * 2);
+    newGraph->adjMat = (unsigned int **)malloc(sizeof(unsigned int *) * numberVertices * 2);
+    unsigned int *temp = (unsigned int *)malloc(sizeof(unsigned int) * numberVertices * 2 * numberVertices * 2);
+    newGraph->size = numberVertices;
+    newGraph->capacity = numberVertices * 2;
+
+    // 配置二维数组
+    for (int i = 0; i < numberVertices * 2; i++) {
+        newGraph->adjMat[i] = temp + i * numberVertices * 2;
+    }
+
+    // 赋值
+    memcpy(newGraph->vertices, vertices, sizeof(int) * numberVertices);
+    for (int i = 0; i < numberVertices; i++) {
+        memcpy(newGraph->adjMat[i], adjMat[i], sizeof(unsigned int) * numberVertices);
+    }
+
+    return newGraph;
+}
+
+/* Driver Code */
+int main() {
+
+    /* 初始化无向图 */
+    int vertices[5] = {1, 3, 2, 5, 4};
+    unsigned int **edge = (unsigned int **)malloc(sizeof(unsigned int *) * 5);
+
+    // 用于构建二维数组的一维指针
+    unsigned int *temp = (unsigned int *)malloc(sizeof(unsigned int) * 25);
+    memset(temp, 0, sizeof(unsigned int) * 25);
+    for (int k = 0; k < 5; k++) {
+        edge[k] = temp + k * 5;
+    }
+
+    // 初始化边
+    edge[0][1] = edge[1][0] = 1;
+    edge[0][3] = edge[3][0] = 1;
+    edge[1][2] = edge[2][1] = 1;
+    edge[2][3] = edge[3][2] = 1;
+    edge[2][4] = edge[4][2] = 1;
+    edge[3][4] = edge[4][3] = 1;
+
+    // 建立无向图
+    graphAdjMat *graph = newGraphic(5, vertices, edge);
+    free(edge);
+    free(temp);
+    printf("\n初始化后，图为:\n");
+    printGraph(graph);
+
+    /* 添加边 */
+    // 顶点 1, 2 的索引分别为 0, 2
+    addEdge(graph, 0, 2);
+    printf("\n添加边 1-2 后图为\n");
+    printGraph(graph);
+
+    /* 删除边 */
+    // 顶点 1, 3 的索引分别为 0, 1
+    removeEdge(graph, 0, 1);
+    printf("\n删除边 1-3 后，图为\n");
+    printGraph(graph);
+
+    /* 添加顶点 */
+    addVertex(graph, 6);
+    printf("\n添加顶点 6 后，图为\n");
+    printGraph(graph);
+
+    /* 删除顶点 */
+    // 顶点 3 的索引为 1
+    removeVertex(graph, 1);
+    printf("\n删除顶点 3 后，图为\n");
+    printGraph(graph);
+
+    return 0;
+}

codes/c/chapter_graph/graph_bfs

@@ -0,0 +1,157 @@
+/**
+ * File: graph_bfs.c
+ * Created Time: 2023-07-11
+ * Author: NI-SW (947743645@qq.com)
+ */
+
+#include "graph_adjacency_list.c"
+
+/* 哈希表 */
+struct hashTable {
+    unsigned int size;
+    unsigned int *array;
+};
+
+typedef struct hashTable hashTable;
+
+/* 初始化哈希表 */
+hashTable *newHash(unsigned int size) {
+    hashTable *h = (hashTable *)malloc(sizeof(hashTable));
+    h->array = (unsigned int *)malloc(sizeof(unsigned int) * size);
+    memset(h->array, 0, sizeof(unsigned int) * size);
+    h->size = size;
+    return h;
+}
+
+/* 标记索引过的顶点 */
+void hashMark(hashTable *h, int index) {
+    h->array[index % h->size] = 1;
+}
+
+/* 查询顶点是否已被标记 */
+int hashQuery(hashTable *h, int index) {
+    // 若顶点已被标记，则返回 0
+    if (h->array[index % h->size] == 1) {
+        return 0;
+    } else {
+        return 1;
+    }
+}
+
+/* 释放哈希表内存 */
+void freeHash(hashTable *h) {
+    free(h->array);
+    free(h);
+}
+
+/* 队列 */
+struct queue {
+    Vertex **list;
+    unsigned int size;
+    int head;
+    int tail;
+};
+
+typedef struct queue queue;
+
+/* 初始化队列 */
+queue *newQueue(unsigned int size) {
+    queue *q = (queue *)malloc(sizeof(queue));
+    q->size = size;
+    q->list = (Vertex **)malloc(sizeof(Vertex *) * size);
+    q->head = 0;
+    q->tail = 0;
+
+    return q;
+}
+
+/* 入队 */
+void queuePush(queue *q, Vertex *v) {
+    q->list[q->tail] = v;
+    q->tail++;
+}
+
+/* 出队 */
+void queuePop(queue *q) {
+    q->head++;
+}
+
+/* 队首元素 */
+Vertex *queueTop(queue *q) {
+    return q->list[q->head];
+}
+
+/* 释放队列内存 */
+void freeQueue(queue *q) {
+    free(q->list);
+    free(q);
+}
+
+/* 广度优先遍历 */
+void graphBFS(graphAdjList *t) {
+    // 初始化队列与哈希表
+    queue *que = newQueue(t->size);
+    hashTable *visited = newHash(t->size);
+    // 将第一个元素入队
+    queuePush(que, t->verticesList[0]);
+    hashMark(visited, t->verticesList[0]->pos);
+
+    printf("\n[");
+    while (que->head < que->tail) {
+        // 遍历该顶点的边链表，将所有与该顶点有连接的，并且未被标记的顶点入队
+        Node *n = queueTop(que)->linked->head->next;
+        while (n != 0) {
+            // 查询哈希表，若该索引的顶点已入队，则跳过，否则入队并标记
+            if (hashQuery(visited, n->val->pos) != 0) {
+                queuePush(que, n->val);
+                hashMark(visited, n->val->pos);
+            }
+            n = n->next;
+        }
+        // 打印队首元素
+        if (que->head == que->tail - 1) {
+            printf("%d]\n", queueTop(que)->val);
+        } else {
+            printf("%d, ", queueTop(que)->val);
+        }
+        // 队首元素出队
+        queuePop(que);
+    }
+    printf("\n");
+
+    // 释放队列与哈希表内存
+    freeQueue(que);
+    freeHash(visited);
+}
+
+int main() {
+
+    /* 初始化无向图 */
+    graphAdjList *graph = newGraphic(3);
+    // 初始化顶点
+    for (int i = 0; i < 10; i++) {
+        addVertex(graph, i);
+    }
+    // 初始化边
+    addEdge(graph, 0, 1);
+    addEdge(graph, 0, 3);
+    addEdge(graph, 1, 2);
+    addEdge(graph, 1, 4);
+    addEdge(graph, 2, 5);
+    addEdge(graph, 3, 4);
+    addEdge(graph, 3, 6);
+    addEdge(graph, 4, 5);
+    addEdge(graph, 4, 7);
+    addEdge(graph, 5, 8);
+    addEdge(graph, 6, 7);
+    addEdge(graph, 7, 8);
+
+    printf("\n初始化后，图为:\n");
+    printGraph(graph);
+    
+    printf("\n广度优先遍历（BFS）顶点序列为");
+    graphBFS(graph);
+
+    return 0;
+}
+