📄 ds7.1.2.htm
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</span>OD(v2)=0<span style="mso-spacerun: yes"> </span>TD(v2)=1</font></b></font></span></p>
<p class="MsoNormal" style="text-indent: 21.0pt; margin-top: 0; margin-bottom: 0"><span lang="EN-US"><span style="mso-spacerun: yes"><font color="#FFFFFF"><b><font size="5">
</font></b></font></span><font color="#FFFFFF"><b><font size="5">ID(v3)=1 <span style="mso-spacerun: yes"> </span><span style="mso-spacerun:
yes"> </span>OD(v3)=1 <span style="mso-spacerun: yes"> </span>TD(v3)=2</font></b></font></span></p>
<p class="MsoNormal" style="text-indent: 21.0pt; margin-top: 0; margin-bottom: 0"><span lang="EN-US"><span style="mso-spacerun: yes"><font color="#FFFFFF"><b><font size="5">
</font></b></font></span><font color="#FFFFFF"><b><font size="5">ID(v4)=1<span style="mso-spacerun: yes">
</span>OD(v4)=1 <span style="mso-spacerun: yes"> </span>TD(v4)=2</font></b></font></span></p>
<p align="left"><font size="5" color="#FFFFFF"><span style="mso-bidi-font-size: 10.0pt; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA"><b>
可以证明，对于具有<span lang="EN-US" style="mso-bidi-font-size: 10.0pt; font-family: Times New Roman; mso-fareast-font-family: 宋体; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">n</span><span style="mso-bidi-font-size: 10.0pt; font-family: 宋体; mso-ascii-font-family: Times New Roman; mso-hansi-font-family: Times New Roman; mso-bidi-font-family: Times New Roman; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">个顶点、</span><span lang="EN-US" style="mso-bidi-font-size: 10.0pt; font-family: Times New Roman; mso-fareast-font-family: 宋体; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">e</span><span style="mso-bidi-font-size: 10.0pt; font-family: 宋体; mso-ascii-font-family: Times New Roman; mso-hansi-font-family: Times New Roman; mso-bidi-font-family: Times New Roman; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">条边的图，顶点</span><span lang="EN-US" style="mso-bidi-font-size: 10.0pt; font-family: Times New Roman; mso-fareast-font-family: 宋体; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">vi</span><span style="mso-bidi-font-size: 10.0pt; font-family: 宋体; mso-ascii-font-family: Times New Roman; mso-hansi-font-family: Times New Roman; mso-bidi-font-family: Times New Roman; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">的度</span><span lang="EN-US" style="mso-bidi-font-size: 10.0pt; font-family: Times New Roman; mso-fareast-font-family: 宋体; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">TD
(vi)</span><span style="mso-bidi-font-size: 10.0pt; font-family: 宋体; mso-ascii-font-family: Times New Roman; mso-hansi-font-family: Times New Roman; mso-bidi-font-family: Times New Roman; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">与顶点的个数以及边的数目满足关系：</span></b></span></font></p>
<p align="center"><b><font color="#FFFFFF" size="5"><span style="mso-bidi-font-size: 10.0pt; font-family: 宋体; mso-ascii-font-family: Times New Roman; mso-hansi-font-family: Times New Roman; mso-bidi-font-family: Times New Roman; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA"> <img border="0" src="ds7.1.3.gif" width="272" height="83">
</span></font></b></p>
<p> </p>
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<p class="MsoNormal"><b><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:
"Times New Roman""><font size="5" color="#FFFF00">边的权、网图</font><font size="5" color="#FFFFFF">:与边有关的数据信息称为</font><font size="5" color="#FFFF00">权</font><font size="5" color="#FFFFFF">（</font></span><font size="5" color="#FFFFFF"><span lang="EN-US">weight</span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:
"Times New Roman"">）。在实际应用中，权值可以有某种含义。比如，在一个反映城市交通线路的图中，边上的权值可以表示该条线路的长度或者等级；对于一个电子线路图，边上的权值可以表示两个端点之间的电阻、电流或电压值；对于反映工程进度的图而言，边上的权值可以表示从前一个工程到后一个工程所需要的时间等等。边上带权的图称为</font><font size="5" color="#FFFF00">网图或网络</font><font size="5" color="#FFFFFF">（</font></span><font size="5" color="#FFFFFF"><span lang="EN-US">network</span><span style="font-family:宋体;mso-ascii-font-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman"">）。如图</span><span lang="EN-US">8.3</span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman"">所示，就是一个无向网图。如果边是有方向的带权图，则就是一个有向网图。</span></font></b><!--mstheme--></font><!--msthemelist--></td>
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<p class="MsoNormal"><b><span style="font-family:
宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman""><font size="5" color="#FFFF00">路径、路径长度</font><font size="5" color="#FFFFFF">:顶点</font></span><font size="5" color="#FFFFFF"><span lang="EN-US">v<sub>p</sub></span><span style="font-family:宋体;mso-ascii-font-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman"">到顶点</span><span lang="EN-US">v<sub>q</sub></span><span style="font-family:宋体;mso-ascii-font-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman"">之间的路径（</span><span lang="EN-US">path</span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman"">）是指顶点序列</span><span lang="EN-US">v<sub>p</sub>,v<sub>i1</sub>,v<sub>i2</sub>,
…, v<sub>im</sub>,v<sub>q.</sub></span><span style="font-family:宋体;mso-ascii-font-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman"">。其中，（</span><span lang="EN-US">v<sub>p</sub>,v<sub>i1</sub></span><span style="font-family:宋体;
mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"">），</span><span lang="EN-US">(v<sub>i1</sub>,v<sub>i2</sub>)</span><span style="font-family:宋体;
mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"">，</span><span lang="EN-US">…,(v<sub>im</sub>,.v<sub>q</sub>)</span><span style="font-family:
宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"">分别为图中的边。路径上边的数目称为路径长度。图</span><span lang="EN-US">8.1</span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman"">所示的无向图</span><span lang="EN-US">G1</span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:
"Times New Roman"">中，</span><span lang="EN-US">v1</span><span style="font-family:
宋体;mso-hansi-font-family:"Times New Roman"">→</span><span lang="EN-US">v4</span><span style="font-family:宋体;mso-hansi-font-family:"Times New Roman"">→</span><span lang="EN-US">v3</span><span style="font-family:宋体;mso-hansi-font-family:"Times New Roman"">→</span><span lang="EN-US">v5</span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman"">与</span><span lang="EN-US">v1</span><span style="font-family:宋体;mso-hansi-font-family:"Times New Roman"">→</span><span lang="EN-US">v2</span><span style="font-family:宋体;mso-hansi-font-family:"Times New Roman"">→</span><span lang="EN-US">v5</span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman"">是从顶点</span><span lang="EN-US">v1
</span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:
"Times New Roman"">到顶点</span><span lang="EN-US">v5 </span><span style="font-family:
宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"">的两条路径，路径长度分别为</span><span lang="EN-US">3</span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman"">和</span><span lang="EN-US">2</span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:
"Times New Roman"">。</span></font></b><!--mstheme--></font><!--msthemelist--></td>
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<p class="MsoNormal"><font size="5" color="#FFFFFF"><b><span style="font-family:
宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"">回路、简单路径、简单回路:称</span><span lang="EN-US">vi</span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman"">的路径为回路或者环（</span><span lang="EN-US">cycle</span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:
"Times New Roman"">）。序列中顶点不重复出现的路径称为简单路径。在图</span><span lang="EN-US">8.1</span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:
"Times New Roman"">中，前面提到的</span><span lang="EN-US">v1</span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:
"Times New Roman"">到</span><span lang="EN-US">v5</span><span style="font-family:
宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"">的两条路径都为简单路径。除第一个顶点与最后一个顶点之外，其他顶点不重复出现的回路称为简单回路，或者简单环。如图</span><span lang="EN-US">8.2</span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman"">中的</span><span lang="EN-US">v1</span><span style="font-family:宋体;mso-hansi-font-family:"Times New Roman"">→</span><span lang="EN-US">v3</span><span style="font-family:宋体;mso-hansi-font-family:"Times New Roman"">→</span><span lang="EN-US">v4</span><span style="font-family:宋体;mso-hansi-font-family:"Times New Roman"">→</span><span lang="EN-US">v1</span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";
mso-hansi-font-family:"Times New Roman"">。</span></b></font><!--mstheme--></font><!--msthemelist--></td>
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<td valign="top" width="100%"><!--mstheme--><font face="宋体"><font size="5" color="#FFFFFF"><b><span style="mso-bidi-font-size: 10.0pt; font-family: 宋体; mso-ascii-font-family: Times New Roman; mso-hansi-font-family: Times New Roman; mso-bidi-font-family: Times New Roman; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">子图:对于图</span><span lang="EN-US" style="mso-bidi-font-size: 10.0pt; font-family: Times New Roman; mso-fareast-font-family: 宋体; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">G=</span><span style="mso-bidi-font-size: 10.0pt; font-family: 宋体; mso-ascii-font-family: Times New Roman; mso-hansi-font-family: Times New Roman; mso-bidi-font-family: Times New Roman; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">（</span><span lang="EN-US" style="mso-bidi-font-size: 10.0pt; font-family: Times New Roman; mso-fareast-font-family: 宋体; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">V</span><span style="mso-bidi-font-size: 10.0pt; font-family: 宋体; mso-ascii-font-family: Times New Roman; mso-hansi-font-family: Times New Roman; mso-bidi-font-family: Times New Roman; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">，</span><span lang="EN-US" style="mso-bidi-font-size: 10.0pt; font-family: Times New Roman; mso-fareast-font-family: 宋体; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">E</span><span style="mso-bidi-font-size: 10.0pt; font-family: 宋体; mso-ascii-font-family: Times New Roman; mso-hansi-font-family: Times New Roman; mso-bidi-font-family: Times New Roman; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">），</span><span lang="EN-US" style="mso-bidi-font-size: 10.0pt; font-family: Times New Roman; mso-fareast-font-family: 宋体; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">G’=</span><span style="mso-bidi-font-size: 10.0pt; font-family: 宋体; mso-ascii-font-family: Times New Roman; mso-hansi-font-family: Times New Roman; mso-bidi-font-family: Times New Roman; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">（</span><span lang="EN-US" style="mso-bidi-font-size: 10.0pt; font-family: Times New Roman; mso-fareast-font-family: 宋体; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">V</span><span style="mso-bidi-font-size: 10.0pt; font-family: 宋体; mso-ascii-font-family: Times New Roman; mso-hansi-font-family: Times New Roman; mso-bidi-font-family: Times New Roman; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">’，</span><span lang="EN-US" style="mso-bidi-font-size: 10.0pt; font-family: Times New Roman; mso-fareast-font-family: 宋体; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">E’</span><span style="mso-bidi-font-size: 10.0pt; font-family: 宋体; mso-ascii-font-family: Times New Roman; mso-hansi-font-family: Times New Roman; mso-bidi-font-family: Times New Roman; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">），若存在</span><span lang="EN-US" style="mso-bidi-font-size: 10.0pt; font-family: Times New Roman; mso-fareast-font-family: 宋体; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">V’</span><span style="mso-bidi-font-size: 10.0pt; font-family: 宋体; mso-ascii-font-family: Times New Roman; mso-hansi-font-family: Times New Roman; mso-bidi-font-family: Times New Roman; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">是</span><span lang="EN-US" style="mso-bidi-font-size: 10.0pt; font-family: Times New Roman; mso-fareast-font-family: 宋体; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">V</span><span style="mso-bidi-font-size: 10.0pt; font-family: 宋体; mso-ascii-font-family: Times New Roman; mso-hansi-font-family: Times New Roman; mso-bidi-font-family: Times New Roman; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">的子集</span><span style="mso-bidi-font-size: 10.0pt; font-family: Times New Roman; mso-fareast-font-family: 宋体; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">
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