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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:

 documentclassstandalone
usepackagetikz
begindocument
begintikzpicture
tikzsetBullet/.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) ;
endtikzpicture
enddocument

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
usepackageetoolbox
usetikzlibrarymatrix,positioning
% building the table in a foreach loop from https://tex.stackexchange.com/a/60400/121799
begindocument
begintikzpicture[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& 1 & 2 & 3 & 4 & 5\
 beginscope[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
 begingroupedefxendgroup
 noexpandgapptonoexpandmymatrixcontentX x
 foreach Y in 1,...,5
 pgfmathtruncatemacroitestadjancymatrix[X-1][Y-1]
 ifnumitest=1
 draw (EX) -- (EY);
 begingroupedefxendgroup
 noexpandgapptonoexpandmymatrixcontent& 1 x
 else
 begingroupedefxendgroup
 noexpandgapptonoexpandmymatrixcontent &x
 fi
 
 gapptomymatrixcontent\
 
 endscope 
 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& 1 & 2 & 3 & 4 & 5\
 beginscope[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
 begingroupedefxendgroup
 noexpandgapptonoexpandmymatrixcontentX x
 foreach Y in 1,...,5
 pgfmathtruncatemacroitestadjancymatrix[X-1][Y-1]
 ifnumitest=1
 draw (EX) -- (EY);
 begingroupedefxendgroup
 noexpandgapptonoexpandmymatrixcontent& 1 x
 else
 begingroupedefxendgroup
 noexpandgapptonoexpandmymatrixcontent &x
 fi
 
 gapptomymatrixcontent\
 
 endscope 
 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& 1 & 2 & 3 & 4 & 5\
 beginscope[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
 begingroupedefxendgroup
 noexpandgapptonoexpandmymatrixcontentX x
 foreach Y in 1,...,5
 pgfmathtruncatemacroitestadjancymatrix[X-1][Y-1]
 ifnumitest=1
 draw (EX) -- (EY);
 begingroupedefxendgroup
 noexpandgapptonoexpandmymatrixcontent& 1 x
 else
 begingroupedefxendgroup
 noexpandgapptonoexpandmymatrixcontent &x
 fi
 
 gapptomymatrixcontent\
 
 endscope 
 matrix (rightmat) [below=of right,adjacency matrix]
 mymatrixcontent
 ;

endtikzpicture
enddocument