带有标签和颜色的内联图形符号

Question 1

我已将解决方案从形状（regular polygon）更改为一些命名的圆形节点。

every corner您可以通过样式（快捷方式corners=<to add>：）和访问节点corner <num>。顶点也是一样：（every vertex）vertices=<to add>和vertex <c1>-<c2>。

还有set corners={<corner list>=<styles>}，set vertices={<vertices list>=<styles>}您可以用它快速设置一个或多个角或顶点的样式，而不必/.append style每次都写入。

使用像宏这样的宏，\triGraph您不必{3}{1-2, 2-3, 3-1}每次都指定。也是如此\biGraph。还有一个\biGraphGround内部绘制具有三个顶点/角的图形，但不绘制第三个（上）角。我们使用它来获得与 s 相同的间距和垂直图像\triGraph。

我还设置了样式，这些样式是填充角圆的快捷方式（x= 填充，o= 无填充）。这样意味着第一个和第三个角将被填充，第二个角将不被填充。对于顶点， = solid 和=也xox存在相同的快捷方式，即，或在下面的示例中。sdd styledasheddensely dasheddensely dotted

这样，第一个矩阵就可以像这样给出

\flaggraphset{size=big,oox}
\begin{pmatrix}
  \triGraph[ddd] + \triGraph[sdd]            
    & \frac {1}{2} (\triGraph[dsd] + \triGraph[ssd]) \\
  \frac {1}{2} (\triGraph[dsd] + \triGraph[ssd])
    & \triGraph[dss]
\end{pmatrix}

为此，我要单独介绍一个鍵盤名称空间（/flaggraph）而不是使用默认值，/tikz这样我们就不会与其他类似的命名样式发生任何冲突。

\tikzset{<kv>}我们现在使用，而不是\flaggraphset{<kv>}。

该样式label corners可用于显示所有弯道的名称，但这只能用于查找某个弯道的编号。

该label 12样式用于标记如图所示的第一个角和第二个角。

代码

\documentclass{article}
\usepackage{tikz}
\usepackage{mathtools,stmaryrd}
\DeclarePairedDelimiter{\downwardoperator}{\llbracket}{\rrbracket_\bullet}
\newcommand*\flaggraphset{\pgfqkeys{/flaggraph}}
\tikzset{
  flag graph/.code=\flaggraphset{#1},
  declare function={angInPolygon(\i,\n)=(\i-1)/\n*360;}}
\flaggraphset{
  tikz/.code=\tikzset{#1},
  vertex path/.style args={#1-#2}{
    insert path={(corner-#1) to (corner-#2)}},
  every corner/.style={
    shape=circle, draw, inner sep=+0pt, minimum size=+1.5pt},
  every vertex/.style=draw,
  %
  size/.is choice,
  d style/.style=densely dashed,
  size/default/.style={every flaggraph/.append style={
    x=.4em, y=.4em, baseline=-.3em, /flaggraph/corners={minimum size=+1.5pt}}},
  size/big/.style={every flaggraph/.append style={
    x=.55em, y=.55em, baseline=-.6ex, /flaggraph/corners={minimum size=+2.5pt}}},
  size/bigger/.style={d style/.style=dashed, every flaggraph/.append style={thick,
    x=2em, y=2em, baseline=-1.25em, /flaggraph/corners={minimum size=+6pt}}},
  size=default,
  %
  vertex 1-2 horizontal/.style={/tikz/rotate=210},
  % setting all corners/vertices
  corners/.style={/flaggraph/every corner/.append style={#1}},
  vertices/.style={/flaggraph/every vertex/.append style={#1}},
  % setting styles
  set corners/.style args={#1=#2}{
    @/.style={corner ##1/.append style={#2}},@/.list={#1}},
  set vertices/.style args={#1=#2}{
    @/.style={vertex ##1/.append style={#2}},@/.list={#1}},
  % debug
  label corners/.style={corners={label={[/flaggraph/label corner]:\Corner}}},
  label corner/.style={overlay,font=\tiny,gray,inner sep=+0pt}
}

\newcommand*\FlagGraph[3][]{
\begin{tikzpicture}[flag graph={#1},/flaggraph/every flaggraph]
  \foreach \Corner in {1,...,#2}
    \node[/flaggraph/every corner, /flaggraph/corner \Corner/.try]
      at ({angInPolygon(\Corner,#2)}:1) (corner-\Corner) {};
  \foreach \Edge in {#3}
    \path[/flaggraph/every vertex, /flaggraph/vertex \Edge/.try,
      /flaggraph/vertex path/.expanded=\Edge];
\end{tikzpicture}}

\newcommand*\biGraphGround[1][]{%
  \FlagGraph[vertex 1-2 horizontal,#1,
    corner 3/.append style={draw=none,fill=none}]{3}{1-2}}
\newcommand*\biGraph[1][]{\FlagGraph[#1]{2}{1-2}}
\flaggraphset{% styles for 2-graphs
  oo/.style={corners={fill=none}},xx/.style={corners=fill},
  xo/.style={corner 1/.append style=fill,corner 2/.append style={fill=none}},
  ox/.style={corner 1/.append style={fill=none},corner 2/.append style=fill},
  s/.style={vertices=solid}, d/.style={vertices=/flaggraph/d style},
  label 12/.style={
    corner 1/.append style={label={[inner sep=1pt, font=\small]left:1}},
    corner 2/.append style={label={[inner sep=1pt, font=\small]right:2}}}
}
\newcommand*\triGraph[1][]{\FlagGraph[vertex 1-2 horizontal,#1]{3}{1-2,2-3,3-1}}
\flaggraphset{% styles for 3-graphs
  ooo/.style={corners={fill=none}},xxx/.style={corners=fill},
  oox/.style={set corners={3=fill},set corners={1,2=fill=none}},
  sss/.style={vertices=solid}, ddd/.style={vertices=/flaggraph/d style},
  sdd/.style={set vertices={2-3,3-1=/flaggraph/d style},set vertices={1-2=solid}},
  dss/.style={set vertices={1-2=/flaggraph/d style},set vertices={2-3,3-1=solid}},
  dsd/.style={set vertices={1-2,3-1=/flaggraph/d style},set vertices={2-3=solid}},
  ssd/.style={set vertices={3-1=/flaggraph/d style},set vertices={1-2,2-3=solid}},
  sds/.style={set vertices={2-3=/flaggraph/d style},set vertices={1-2,3-1=solid}},
}
% o = don't fill corner, x = fill corner
% s = solid line,        d = dashed line
\begin{document}
Text: \FlagGraph[set corners={3,5 = red}]{5}{1-2,1-4}\par
Text: \FlagGraph[set corners={2 = blue}]{3}{1-2}\par
Text: \FlagGraph[vertices=bend left]{2}{1-2}\par
Text: \FlagGraph[label corners]{2}{}\par
Text: \FlagGraph[corners=green,
  set vertices={3-1=densely dotted}]{5}{3-1,3-2,3-4,3-5}\par

\[ vv^\top = \flaggraphset{size=big,oox}
\begin{pmatrix}
  \triGraph[ddd] + \triGraph[sdd]            
    & \frac 12 (\triGraph[dsd] + \triGraph[ssd]) \\
  \frac 12 (\triGraph[dsd] + \triGraph[ssd]) & \triGraph[dss]
\end{pmatrix}
\]
and then apply the downward operator to get
\[ \downwardoperator{vv^\top} = \flaggraphset{size=big}
\begin{pmatrix}
  \triGraph[ddd] + \frac 13 \triGraph[sdd]   
    & \frac 13 (\triGraph[sdd] + \triGraph[dss]) \\
  \frac 13 (\triGraph[sdd] + \triGraph[dss]) & \triGraph[dss]
\end{pmatrix}
\]

\begin{center}
\flaggraphset{size=bigger}
\begin{tabular}{*{5}{c}}
  \biGraphGround[d] &    \biGraphGround[s]     &      \triGraph[ddd]      &     \triGraph[sdd]      & \triGraph[dss] \\
                    & \triGraph[label 12, sdd] & \triGraph[label 12, sds] & \triGraph[label 12,ssd] &
\end{tabular}
\end{center}

% Instead of \textbullet, something like this could be used:
% \FlagGraph[size=big,corners=fill]{1}{}
\flaggraphset{size=big, d style/.style=densely dotted}
These are the \textbullet-flags of sizes 2 and 3:
\biGraph[xo,d], \biGraph[xo], {\flaggraphset{oox}\triGraph[ddd], \triGraph[sdd],
  \triGraph[dsd], \triGraph[dss], and \triGraph[ssd].
\end{document}

输出

Answer