圖的基本性質

圖的分類：圖可以分為有向圖和無向圖，如圖為無向圖：

另外，還可以將圖分為有權圖和無權圖，權表示兩個節點之間的某種關係，比如交通運輸網路中兩個地點的權值可以代表交通費用，如圖為有向圖：

連通性：有時候，可能有兩個區域的交通路線是沒有相連的，這就會產生很多不連通的區域，如圖就是不完全連通的：

簡單圖

圖的表示方法

鄰接矩陣
如圖為一個無向圖，要用具體的數值表示節點之間的關係，可以使用鄰接矩陣，假設這個矩陣是A,Aij$A_{ij}$就表示第i個節點和第j個節點是否相連，為1表示相連，0表示不相連。

除此之外，還可以使用鄰接矩陣表示有向圖：

鄰接表
用鄰接矩陣來表示，每一行表示一個節點與其他所有節點是否相連，但對於鄰接表來說，一行只代表和他相連的節點：

圖的python實現

鄰接矩陣

class DenseGraph:
    def __init__(self,n,directed = False):
        self.n = n # number of vertex
        self.m = 0 #number of edge
        self.directed = directed
        self.matrix = [[0 for i in  range(n)] for i in range(n)]

    def
 
 __str__(self):
        for line in self.matrix:
            print(str(line))
        return '' # must return string

    def getNumberOfEdge(self):
        return self.m
    def getNumberOfVertex(self):
        return self.n

    def hasEdge(self,v,w):
        if 0 <= v <= self.n and 0 <= w <= self.n:
            return
 
 self.matrix[v][w]
        else:
            raise Exception("vertex not in the Graph")

    def addEdge(self,v,w):
        if 0 <= v <= self.n and 0 <= w <= self.n:
            if self.hasEdge(v,w):
                return
            self.matrix[v][w]= 1
            if self.directed is False:
                self.matrix[w][v] = 1
            self.m += 1
        else:
            raise Exception("vertex not in the Graph")

鄰接表

lass Vertex(object):
    def __init__(self,key):
        self.id = key
        self.connectedTo = {} #the key is vertex,value is weight

    def addNeighbor(self, nbr, weight=0):
        self.connectedTo[nbr] = weight

    def __str__(self):
        return str(self.id) + ' connectedTo: ' + str([x.id for x in self.connectedTo])

    def getConnections(self):
        return self.connectedTo.keys()

    def getConnectionsId(self):
        idList = []
        for k in self.connectedTo.keys():
            idList.append(k.getId())
        return sorted(idList)

    def getConnectionsIdAndWeight(self):
        idList = []
        for k in self.connectedTo.keys():
            idList.append(k.getId())
        weightList = list(self.connectedTo.values())
        return {idList[i]: weightList[i] for i in range(len(idList))}

    def getWeight(self, nbr):
        return self.connectedTo[nbr]

    def getId(self):
        return self.id


class SparseGraph(object):
    def __init__(self,directed=False,weighted=False):
        self.vertDict = {} #key is the id of vertex，value is vertex
        self.numVertices = 0
        self.directed=directed
        self.weighted=weighted

    def addVertex(self,key):
        self.numVertices = self.numVertices + 1
        newVertex = Vertex(key)
        self.vertDict[key] = newVertex
        return newVertex

    def getVertex(self,n):
        if n in self.vertDict:
            return self.vertDict[n]
        else:
            return None

    def __contains__(self,n):
        return n in self.vertDict

    def addEdge(self,f,t,weight=0):
        if f not in self.vertDict:
            self.addVertex(f)
        if t not in self.vertDict:
            self.addVertex(t)
        self.vertDict[f].addNeighbor(self.vertDict[t], weight)
        if self.directed is False:
            self.vertDict[t].addNeighbor(self.vertDict[f], weight)

    def getVertices(self):
        return list(self.vertDict.keys())

    def getVertNum(self):
        return self.numVertices

    def __iter__(self):
        return iter(self.vertDict.values())

    def getAllInfo(self):
        verticesList=[int(x) for x in list(self.getVertices())]
        verticesList.sort()
        if self.weighted:
            for i in range(len(verticesList)):
                print('vertex %s : %s' % (i, self.getVertex(i).getConnectionsIdAndWeight()))
        else:
            for i in range(len(verticesList)):
                print('vertex %s : %s' %(i,self.getVertex(i).getConnectionsId()))

打印出來看一下：

def buildGraphFromFile(aGraph,filePath):
    graphList=[]
    with open(filePath,'r',encoding='utf-8') as f:
        for line in f:
            graphList.append([int(x) for x in re.split(r'\s+',line.strip())])
    for i in range(len(graphList)):
        aGraph.addEdge(graphList[i][0],graphList[i][1])


# g1=DenseGraph(13)  #必須填入正確的結點個數。。。我真的覺得鄰接矩陣不好用
# buildGraphFromFile(g1,'/Users/huanghuaixian/desktop/testG1.txt')
# print(g1)


g2=SparseGraph()
buildGraphFromFile(g2,'/Users/huanghuaixian/desktop/testG2.txt')
g2.getAllInfo()