Ggthjy

Here are some isomorphic graphs and their corresponding adjacency matrices. I can draw the graphs with tikz. But I'm not sure the best way to draw the matrices. Is it possible to generate one from the other? What's the right way to approach it?

Here's an example of the code to generate a graph:

 documentclass{standalone}

usepackage{tikz}

begin{document}

begin{tikzpicture}

tikzset{Bullet/.style={circle,draw,fill=black,scale=0.75}}

node[Bullet,label=left :{$e_1$}] (E1) at (0,2) {} ;

node[Bullet,label=above:{$e_2$}] (E2) at (1,3) {} ;

node[Bullet,label=right:{$e_3$}] (E3) at (2,2) {} ;

node[Bullet,label=right:{$e_4$}] (E4) at (2,0) {} ;

node[Bullet,label=left :{$e_5$}] (E5) at (0,0) {} ;

draw[thick] (E1)--(E2)--(E3)--(E4)--(E5)--(E1) {} ;

end{tikzpicture}

end{document}

This is in case you change your mind and use the adjacency matrices to draw the graphs. TikZ allows you to define arrays, see p. 999 of the pgfmanual. And these arrays can be converted to tables using this nice answer. And these matrices/arrays can also be used to define the graphs.

documentclass[tikz,border=3.14mm]{standalone}

usepackage{etoolbox}

usetikzlibrary{matrix,positioning}

% building the table in a foreach loop from https://tex.stackexchange.com/a/60400/121799

begin{document}

begin{tikzpicture}[Bullet/.style={circle,draw,fill=black,inner sep=1.5pt},

adjacency matrix/.style={ampersand replacement=&,matrix of math nodes,

 row 1/.append style={nodes={font=boldmath}},

 column 1/.append style={nodes={font=boldmath}},nodes in empty cells,

nodes={draw,minimum width=1.5em,text height=1.8ex},column sep=-pgflinewidth,row

sep=-pgflinewidth}]

% first matrix

defadjancymatrix{%

{{0,0,1,1,0},%

{0,0,0,1,1},%

{1,0,0,0,1},%

{1,1,0,0,0},%

{0,1,1,0,0}}} 

 letmymatrixcontentempty

 defmymatrixcontent{|[draw=none]|& 1 & 2 & 3 & 4  & 5\}

 begin{scope}[local bounding box=left]

  foreach X in {1,...,5}

   {node[Bullet,label=90+72-X*72:{$e_X$}] (EX) at (90+72-X*72:2) {} ;}

  foreach X in {1,...,5}

  {begingroupedefx{endgroup

         noexpandgapptonoexpandmymatrixcontent{X }}x

  foreach Y in {1,...,5}

   {pgfmathtruncatemacro{itest}{adjancymatrix[X-1][Y-1]}

   ifnumitest=1

    draw (EX) -- (EY);

    begingroupedefx{endgroup

         noexpandgapptonoexpandmymatrixcontent{& 1 }}x

   else

    begingroupedefx{endgroup

         noexpandgapptonoexpandmymatrixcontent{ &}}x

   fi

   }

   gapptomymatrixcontent{\}

   }

 end{scope} 

 matrix (leftmat) [below=of left,adjacency matrix]{

    mymatrixcontent

  };

 %

% second matrix

defadjancymatrix{%

{{0,1,0,0,1},%

{1,0,1,0,0},%

{0,1,0,1,0},%

{0,0,1,0,1},%

{1,0,0,1,0}}} 

 letmymatrixcontentempty

 defmymatrixcontent{|[draw=none]|& 1 & 2 & 3 & 4  & 5\}

 begin{scope}[local bounding box=middle,xshift=5cm]

  foreach X in {1,...,5}

   {node[Bullet,label=90+72-X*72:{$e_X$}] (EX) at (90+72-X*72:2) {} ;}

  foreach X in {1,...,5}

  {begingroupedefx{endgroup

         noexpandgapptonoexpandmymatrixcontent{X }}x

  foreach Y in {1,...,5}

   {pgfmathtruncatemacro{itest}{adjancymatrix[X-1][Y-1]}

   ifnumitest=1

    draw (EX) -- (EY);

    begingroupedefx{endgroup

         noexpandgapptonoexpandmymatrixcontent{& 1 }}x

   else

    begingroupedefx{endgroup

         noexpandgapptonoexpandmymatrixcontent{ &}}x

   fi

   }

   gapptomymatrixcontent{\}

   }

 end{scope} 

 matrix (midmat) [below=of middle,adjacency matrix]{

    mymatrixcontent

  };

% third matrix

defadjancymatrix{%

{{0,1,0,1,0},%

{1,0,0,0,1},%

{0,0,0,1,1},%

{1,0,1,0,0},%

{0,1,1,0,0}}} 

 letmymatrixcontentempty

 defmymatrixcontent{|[draw=none]|& 1 & 2 & 3 & 4  & 5\}

 begin{scope}[local bounding box=right,xshift=10cm]

  foreach X in {1,...,3}

   {node[Bullet,label=90+72-X*72:{$e_X$}] (EX) at (90+72-X*72:2) {} ;}

  node[Bullet,label=90+72-4*72:{$e_5$}] (E5) at (90+72-4*72:2) {} ; 

  node[Bullet,label=90+72-5*72:{$e_4$}] (E4) at (90+72-5*72:2) {} ; 

  foreach X in {1,...,5}

  {begingroupedefx{endgroup

         noexpandgapptonoexpandmymatrixcontent{X }}x

  foreach Y in {1,...,5}

   {pgfmathtruncatemacro{itest}{adjancymatrix[X-1][Y-1]}

   ifnumitest=1

    draw (EX) -- (EY);

    begingroupedefx{endgroup

         noexpandgapptonoexpandmymatrixcontent{& 1 }}x

   else

    begingroupedefx{endgroup

         noexpandgapptonoexpandmymatrixcontent{ &}}x

   fi

   }

   gapptomymatrixcontent{\}

   }

 end{scope} 

 matrix (rightmat) [below=of right,adjacency matrix]{

    mymatrixcontent

  };



end{tikzpicture}

end{document}

Here is a sagetex solution which uses the computer algebra system, SAGE, to do the work. SAGE has built in knowledge of different classes of graphs and has some compatibility with LaTeX and Tikz and can solve some graph parameters as well. All that knowledge means SAGE is not part of the LaTeX distribution, but this is easily handled with a free Cocalc account. It might be my lack of knowledge about the subject but I had trouble trying to get the graphs to look exactly the way you have drawn them but if you're willing to give up some control here is a straightforward implementation of what you want.

documentclass{article}

usepackage{sagetex}

usepackage{tikz,tkz-graph,tkz-berge}

begin{document}

begin{sagesilent}

H = Graph({1:[2], 2:[3], 3:[4], 4:[5], 5:[1]})

H.set_pos(H.layout_circular()) #arrange the vertices in a circle

H.set_latex_options(scale=1.0,graphic_size=(3,3))

######

J = Graph({1:[2], 2:[5], 3:[4], 4:[1], 5:[3]})

J.set_pos(H.layout_circular()) #arrange the vertices in a circle

J.set_latex_options(scale=1.0,graphic_size=(3,3))

######

K = Graph({1:[3], 2:[4], 3:[5], 4:[1], 5:[2]})

K.set_pos(H.layout_circular()) #arrange the vertices in a circle

K.set_latex_options(scale=1.0,graphic_size=(3,3))

end{sagesilent}

Consider the three graphs below:\\



begin{tabular}{ccc}

begin{tikzpicture}

GraphInit[vstyle=Normal]

SetVertexNormal[Shape=circle,LineWidth = 1pt]

tikzset{EdgeStyle/.append style = {color = black, line width=1pt}}

sage{H} 

end{tikzpicture} &

begin{tikzpicture}

GraphInit[vstyle=Normal]

SetVertexNormal[Shape=circle,LineWidth = 1pt]

tikzset{EdgeStyle/.append style = {color = black, line width=1pt}}

sage{J} 

end{tikzpicture} & 

begin{tikzpicture}

GraphInit[vstyle=Normal]

SetVertexNormal[Shape=circle,LineWidth = 1pt]

tikzset{EdgeStyle/.append style = {color = black, line width=1pt}}

sage{K} 

end{tikzpicture}\

$sage{H.adjacency_matrix()}$ & $sage{J.adjacency_matrix()}$ &    $sage{K.adjacency_matrix()}$\\

(a) & (b) & (c)\

end{tabular}

end{document}

The output is:

I set up the graph as a list of what a vertex is adjacent to; eg, H = Graph({1:[2], 2:[3], 3:[4], 4:[5], 5:1}) says that vertex 1 is adjacent to 2, vertex 2 is adjacent to 3, and so until vertex 5 is adjacent to 1. Setting a circular layout makes Sage place the vertices for you, and by setting options we can change the size of the graphic. So Sage creates the graph with sage{H} and the adjacency matrix with $sage{H.adjacency_matrix()}$.

By having a CAS do the work you can change the graph and SAGE will do the work without mistakes. This saves you having to code other examples and possibly making a mistake along the way.