G1 := Graph({["A", "B"], ["A", "D"], ["B", "C"], ["C", "E"], ["D", "E"]});

$\mathrm{G1}≔\mathrm{Graph\; 1:\; a\; directed\; graph\; with\; 5\; vertices\; and\; 5\; arcs}$

DrawGraph(G1);

iterator := AllSimplePaths( G1, "A", "E" );

$\mathrm{iterator}≔\left[\begin{array}{c}\mathrm{Path\; Iterator}\end{array}\right]$

iterator:-getNext();

$\left[''A''\,''D''\,''E''\right]$

iterator:-getNext();

$\left[''A''\,''B''\,''C''\,''E''\right]$

iterator:-hasNext();

$\mathrm{false}$

Ancestors( G1, "A" );

Ancestors( G1, "E" );

$\left[''A''\,''B''\,''C''\,''D''\right]$

Descendants( G1, "A" );

$\left[''B''\,''C''\,''D''\,''E''\right]$

FindCycle(G1);

FindCycle( Graph( {["A", "B"], ["B", "C"], ["C", "A"]} ) );

$\left[''C''\,''A''\,''B''\,''C''\right]$

CT := Graph({{1,4},{2,4},{3,4},{4,5},{5,6},{6,7},{7,8},{7,9}});

$\mathrm{CT}≔\mathrm{Graph\; 2:\; an\; undirected\; graph\; with\; 9\; vertices\; and\; 8\; edges}$

DrawGraph(CT);

IsCaterpillarTree(CT);

$\mathrm{true}$

LT := Graph({{1,2},{2,3},{3,4},{4,5},{3,6},{6,7},{3,8},{8,9}});

$\mathrm{LT}≔\mathrm{Graph\; 3:\; an\; undirected\; graph\; with\; 9\; vertices\; and\; 8\; edges}$

DrawGraph(LT);

IsLobsterTree(LT);

$\mathrm{true}$

IsCaterpillarTree(LT);

$\mathrm{false}$

IsPlatonicGraph( SpecialGraphs:-CubeGraph() );

$\mathrm{true}$

LongestPath(G1);

$\left[''A''\,''B''\,''C''\,''E''\right]$

LowestCommonAncestors( G1, "C", "D" );

$\left\{''A''\right\}$

ModularityMatrix(G1);

$\left[\begin{array}{ccccc}0& 1& 0& 1& 0\\ -\frac{2}{5}& -\frac{1}{5}& \frac{4}{5}& -\frac{1}{5}& 0\\ -\frac{2}{5}& -\frac{1}{5}& -\frac{1}{5}& -\frac{1}{5}& 1\\ -\frac{2}{5}& -\frac{1}{5}& -\frac{1}{5}& -\frac{1}{5}& 1\\ -\frac{4}{5}& -\frac{2}{5}& -\frac{2}{5}& -\frac{2}{5}& 0\end{array}\right]$

ResistanceDistance(SpecialGraphs:-CubeGraph());

$\left[\begin{array}{cccccccc}0& \frac{7}{12}& \frac{7}{12}& \frac{3}{4}& \frac{7}{12}& \frac{3}{4}& \frac{3}{4}& \frac{5}{6}\\ \frac{7}{12}& 0& \frac{3}{4}& \frac{7}{12}& \frac{3}{4}& \frac{7}{12}& \frac{5}{6}& \frac{3}{4}\\ \frac{7}{12}& \frac{3}{4}& 0& \frac{7}{12}& \frac{3}{4}& \frac{5}{6}& \frac{7}{12}& \frac{3}{4}\\ \frac{3}{4}& \frac{7}{12}& \frac{7}{12}& 0& \frac{5}{6}& \frac{3}{4}& \frac{3}{4}& \frac{7}{12}\\ \frac{7}{12}& \frac{3}{4}& \frac{3}{4}& \frac{5}{6}& 0& \frac{7}{12}& \frac{7}{12}& \frac{3}{4}\\ \frac{3}{4}& \frac{7}{12}& \frac{5}{6}& \frac{3}{4}& \frac{7}{12}& 0& \frac{3}{4}& \frac{7}{12}\\ \frac{3}{4}& \frac{5}{6}& \frac{7}{12}& \frac{3}{4}& \frac{7}{12}& \frac{3}{4}& 0& \frac{7}{12}\\ \frac{5}{6}& \frac{3}{4}& \frac{3}{4}& \frac{7}{12}& \frac{3}{4}& \frac{7}{12}& \frac{7}{12}& 0\end{array}\right]$

ShortestAncestralPath( G1, "C", "D") ;

$\left[\left[''A''\,''B''\,''C''\right]\,\left[''A''\,''D''\right]\right]$

IsReachable( G1, "A", "E", distance = 1 );

$\mathrm{false}$

Reachable( G1, "A", distance = 1 );

$\left[''A''\,''B''\,''D''\right]$

ShortestPath( G1, "A", "E" );

$\left[''A''\,''D''\,''E''\right]$

ShortestPath( G1, "A", "E", avoidvertices = {"D"} );

$\left[''A''\,''B''\,''C''\,''E''\right]$