$G:=\mathrm{Graph}\left(\left\{\left\{1\,2\right\}\,\left\{2\,3\right\}\,\left\{3\,4\right\}\right\}\right)$

$G≔\mathrm{Graph\; 7:\; an\; undirected\; unweighted\; graph\; with\; 4\; vertices\; and\; 3\; edge(s)}$

$\mathrm{Vertices}\left(G\right)$

$\left[1\,2\,3\,4\right]$

$\mathrm{AdjacencyMatrix}\left(G\right)$

$\left[\begin{array}{rrrr}0& 1& 0& 0\\ 1& 0& 1& 0\\ 0& 1& 0& 1\\ 0& 0& 1& 0\end{array}\right]$

$H:=\mathrm{Digraph}\left(\left\{\left[1\,2\right]\,\left[2\,3\right]\,\left[3\,4\right]\right\}\right)$

$H≔\mathrm{Graph\; 8:\; a\; directed\; unweighted\; graph\; with\; 4\; vertices\; and\; 3\; arc(s)}$

$\mathrm{AdjacencyMatrix}\left(H\right)$

$\left[\begin{array}{rrrr}0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 0\end{array}\right]$

$A:=\left[\begin{array}{rrrr}0& 1& 0& 0\\ 1& 0& 1& 0\\ 0& 1& 0& 1\\ 0& 0& 1& 0\end{array}\right]$

$A≔\left[\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 1& 0\\ 0& 1& 0& 1\\ 0& 0& 1& 0\end{array}\right]$

$G:=\mathrm{Graph}\left(A\right)$

$G≔\mathrm{Graph\; 9:\; an\; undirected\; unweighted\; graph\; with\; 4\; vertices\; and\; 3\; edge(s)}$

$\mathrm{Vertices}\left(G\right)$

$\left[1\,2\,3\,4\right]$

$\mathrm{Edges}\left(G\right)$

$\left\{\left\{1\,2\right\}\,\left\{2\,3\right\}\,\left\{3\,4\right\}\right\}$

$B:=\left[\begin{array}{rrrr}0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 0\end{array}\right]$

$B≔\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 0\end{array}\right]$

$H:=\mathrm{Digraph}\left(\left[a\,b\,c\,d\right]\,B\right)$

$H≔\mathrm{Graph\; 10:\; a\; directed\; unweighted\; graph\; with\; 4\; vertices\; and\; 3\; arc(s)}$

$\mathrm{Vertices}\left(H\right)$

$\left[a\,b\,c\,d\right]$

$\mathrm{Edges}\left(H\right)$

$\left\{\left[a\,b\right]\,\left[b\,c\right]\,\left[c\,d\right]\right\}$

$L:=\left[\left[2\right]\,\left[1\,3\right]\,\left[2\,4\right]\,\left[3\right]\right]$

$L≔\left[\left[2\right]\,\left[1\,3\right]\,\left[2\,4\right]\,\left[3\right]\right]$

$G:=\mathrm{Graph}\left(L\right)$

$G≔\mathrm{Graph\; 11:\; an\; undirected\; unweighted\; graph\; with\; 4\; vertices\; and\; 3\; edge(s)}$

$\mathrm{Edges}\left(G\right)$

$\left\{\left\{1\,2\right\}\,\left\{2\,3\right\}\,\left\{3\,4\right\}\right\}$

$\mathrm{Neighbors}\left(G\right)$

$\left[\left[2\right]\,\left[1\,3\right]\,\left[2\,4\right]\,\left[3\right]\right]$

$L:=\left[\left[2\right]\,\left[3\right]\,\left[4\right]\,\left[1\right]\right]$

$L≔\left[\left[2\right]\,\left[3\right]\,\left[4\right]\,\left[1\right]\right]$

$H:=\mathrm{Digraph}\left(L\right)$

$H≔\mathrm{Graph\; 12:\; a\; directed\; unweighted\; graph\; with\; 4\; vertices\; and\; 4\; arc(s)}$

$\mathrm{Neighbors}\left(H\right)$

$\left[\left[2\,4\right]\,\left[1\,3\right]\,\left[2\,4\right]\,\left[1\,3\right]\right]$

$\mathrm{Arrivals}\left(H\right)$

$\left[\left[4\right]\,\left[1\right]\,\left[2\right]\,\left[3\right]\right]$

$\mathrm{Departures}\left(H\right)$

$\left[\left[2\right]\,\left[3\right]\,\left[4\right]\,\left[1\right]\right]$

$T:=\mathrm{Trail}\left(1\,2\,3\,4\,1\right)$

$T≔\mathrm{Trail}\left(1\,2\,3\,4\,1\right)$

$G:=\mathrm{Graph}\left(T\right)$

$G≔\mathrm{Graph\; 13:\; an\; undirected\; unweighted\; graph\; with\; 4\; vertices\; and\; 4\; edge(s)}$

$\mathrm{Edges}\left(G\right)$

$\left\{\left\{1\,2\right\}\,\left\{1\,4\right\}\,\left\{2\,3\right\}\,\left\{3\,4\right\}\right\}$

$H:=\mathrm{Digraph}\left(\mathrm{Trail}\left(a\,b\,c\,d\,c\,b\,a\right)\right)$

$H≔\mathrm{Graph\; 14:\; a\; directed\; unweighted\; graph\; with\; 4\; vertices\; and\; 6\; arc(s)}$

$\mathrm{Edges}\left(H\right)$

$\left\{\left[a\,b\right]\,\left[b\,a\right]\,\left[b\,c\right]\,\left[c\,b\right]\,\left[c\,d\right]\,\left[d\,c\right]\right\}$

$G:=\mathrm{Graph}\left(5\,\mathrm{weighted}\right)$

$G≔\mathrm{Graph\; 15:\; an\; undirected\; weighted\; graph\; with\; 5\; vertices\; and\; 0\; edge(s)}$

$\mathrm{AddEdge}\left(G\,\left[\left\{1\,2\right\}\,2.3\right]\right)\;$$\mathrm{AddEdge}\left(G\,\left[\left\{3\,4\right\}\,4.5\right]\right)$

$\mathrm{Graph\; 15:\; an\; undirected\; weighted\; graph\; with\; 5\; vertices\; and\; 1\; edge(s)}$

$\mathrm{Graph\; 15:\; an\; undirected\; weighted\; graph\; with\; 5\; vertices\; and\; 2\; edge(s)}$

$\mathrm{AdjacencyMatrix}\left(G\right)$

$\left[\begin{array}{rrrrr}0& 1& 0& 0& 0\\ 1& 0& 0& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right]$

$W:=\mathrm{WeightMatrix}\left(G\right)$

$W≔\left[\begin{array}{ccccc}0& 2.3& 0& 0& 0\\ 2.3& 0& 0& 0& 0\\ 0& 0& 0& 4.5& 0\\ 0& 0& 4.5& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right]$

$G:=\mathrm{Graph}\left(5\,\left\{\left[\left\{1\,2\right\}\,2.3\right]\,\left[\left\{3\,4\right\}\,4.5\right]\right\}\right)$

$G≔\mathrm{Graph\; 16:\; an\; undirected\; weighted\; graph\; with\; 5\; vertices\; and\; 2\; edge(s)}$

$\mathrm{AddEdge}\left(G\,\left[\left\{2\,3\right\}\,0.0\right]\right)$

$\mathrm{Graph\; 16:\; an\; undirected\; weighted\; graph\; with\; 5\; vertices\; and\; 3\; edge(s)}$

$\mathrm{Edges}\left(G\right)$

$\left\{\left\{1\,2\right\}\,\left\{2\,3\right\}\,\left\{3\,4\right\}\right\}$

$\mathrm{AdjacencyMatrix}\left(G\right)$

$\left[\begin{array}{rrrrr}0& 1& 0& 0& 0\\ 1& 0& 1& 0& 0\\ 0& 1& 0& 1& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right]$

$\mathrm{WeightMatrix}\left(G\right)$

$\left[\begin{array}{ccccc}0& 2.3& 0& 0& 0\\ 2.3& 0& 0.& 0& 0\\ 0& 0.& 0& 4.5& 0\\ 0& 0& 4.5& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right]$

$P:=\mathrm{PathGraph}\left(4\right)$

$P≔\mathrm{Graph\; 17:\; an\; undirected\; unweighted\; graph\; with\; 4\; vertices\; and\; 3\; edge(s)}$

$\mathrm{Edges}\left(P\right)$

$\left\{\left\{1\,2\right\}\,\left\{2\,3\right\}\,\left\{3\,4\right\}\right\}$

$C:=\mathrm{CycleGraph}\left(4\right)$

$C≔\mathrm{Graph\; 18:\; an\; undirected\; unweighted\; graph\; with\; 4\; vertices\; and\; 4\; edge(s)}$

$\mathrm{Edges}\left(C\right)$

$\left\{\left\{1\,2\right\}\,\left\{1\,4\right\}\,\left\{2\,3\right\}\,\left\{3\,4\right\}\right\}$

$\mathrm{K4}:=\mathrm{CompleteGraph}\left(4\right)$

$\mathrm{K4}≔\mathrm{Graph\; 19:\; an\; undirected\; unweighted\; graph\; with\; 4\; vertices\; and\; 6\; edge(s)}$

$\mathrm{Edges}\left(\mathrm{K4}\right)$

$\left\{\left\{1\,2\right\}\,\left\{1\,3\right\}\,\left\{1\,4\right\}\,\left\{2\,3\right\}\,\left\{2\,4\right\}\,\left\{3\,4\right\}\right\}$

$\mathrm{IsPlanar}\left(\mathrm{K4}\right)$

$\mathrm{true}$

$\mathrm{K33}:=\mathrm{CompleteGraph}\left(3\,3\right)$

$\mathrm{K33}≔\mathrm{Graph\; 20:\; an\; undirected\; unweighted\; graph\; with\; 6\; vertices\; and\; 9\; edge(s)}$

$\mathrm{IsPlanar}\left(\mathrm{K33}\right)$

$\mathrm{false}$

$G:=\mathrm{Graph}\left(\left\{\left\{1\,2\right\}\,\left\{2\,3\right\}\,\left\{3\,1\right\}\,\left\{3\,4\right\}\,\left\{4\,5\right\}\,\left\{5\,6\right\}\right\}\right)$

$G≔\mathrm{Graph\; 26:\; an\; undirected\; unweighted\; graph\; with\; 6\; vertices\; and\; 6\; edge(s)}$

$\mathrm{DrawGraph}\left(G\right)$

$\mathrm{IsConnected}\left(G\right)$

$\mathrm{true}$

$\mathrm{ConnectedComponents}\left(G\right)$

$\left[\left[1\,2\,3\,4\,5\,6\right]\right]$

$\mathrm{IsBiconnected}\left(G\right)$

$\mathrm{false}$

$\mathrm{BiconnectedComponents}\left(G\right)$

$\left[\left[5\,6\right]\,\left[4\,5\right]\,\left[3\,4\right]\,\left[1\,2\,3\right]\right]$

$\mathrm{AdjacencyMatrix}\left(G\right)$

$\left[\begin{array}{rrrrrr}0& 1& 1& 0& 0& 0\\ 1& 0& 1& 0& 0& 0\\ 1& 1& 0& 1& 0& 0\\ 0& 0& 1& 0& 1& 0\\ 0& 0& 0& 1& 0& 1\\ 0& 0& 0& 0& 1& 0\end{array}\right]$

$\mathrm{Distance}\left(G\,1\,6\right)$

$4$

$\mathrm{AllPairsDistance}\left(G\right)$

$\left[\begin{array}{rrrrrr}0& 1& 1& 2& 3& 4\\ 1& 0& 1& 2& 3& 4\\ 1& 1& 0& 1& 2& 3\\ 2& 2& 1& 0& 1& 2\\ 3& 3& 2& 1& 0& 1\\ 4& 4& 3& 2& 1& 0\end{array}\right]$

$\mathrm{ArticulationPoints}\left(G\right)$

$\left[3\,4\,5\right]$

$H:=\mathrm{DeleteVertex}\left(G\,3\right)$

$H≔\mathrm{Graph\; 27:\; an\; undirected\; unweighted\; graph\; with\; 5\; vertices\; and\; 3\; edge(s)}$

$\mathrm{IsConnected}\left(H\right)$

$\mathrm{false}$

$\mathrm{ConnectedComponents}\left(H\right)$

$\left[\left[1\,2\right]\,\left[4\,5\,6\right]\right]$

$\mathrm{DrawGraph}\left(H\right)$

$H:=\mathrm{InducedSubgraph}\left(G\,\left[3\,4\,5\right]\right)$

$H≔\mathrm{Graph\; 28:\; an\; undirected\; unweighted\; graph\; with\; 3\; vertices\; and\; 2\; edge(s)}$

$\mathrm{Vertices}\left(H\right)$

$\left[3\,4\,5\right]$

$P:=\mathrm{SpecialGraphs}\[\mathrm{PetersenGraph}\]\left(\right)$

$P≔\mathrm{Graph\; 29:\; an\; undirected\; unweighted\; graph\; with\; 10\; vertices\; and\; 15\; edge(s)}$

$\mathrm{DrawGraph}\left(P\right)$

$\mathrm{IsRegular}\left(P\right)$

$\mathrm{true}$

$\mathrm{DegreeSequence}\left(P\right)$

$\left[3\,3\,3\,3\,3\,3\,3\,3\,3\,3\right]$

$\mathrm{CliqueNumber}\left(P\right)$

$2$

$\mathrm{IsPlanar}\left(P\right)$

$\mathrm{false}$

$\mathrm{IsHamiltonian}\left(P\right)$

$\mathrm{false}$

$\mathrm{Girth}\left(P\right)$

$5$

$\mathrm{ChromaticNumber}\left(P\right)$

$3$

$M:=\mathrm{Digraph}\left(\left\{\left[1\,2\right]\,\left[2\,1\right]\,\left[1\,3\right]\,\left[2\,4\right]\,\left[2\,5\right]\,\left[4\,5\right]\,\left[5\,4\right]\right\}\right)$

$M≔\mathrm{Graph\; 30:\; a\; directed\; unweighted\; graph\; with\; 5\; vertices\; and\; 7\; arc(s)}$

$\mathrm{DrawGraph}\left(M\right)$

$\mathrm{IsConnected}\left(M\right)$

$\mathrm{true}$

$\mathrm{IsAcyclic}\left(M\right)$

$\mathrm{false}$

$\mathrm{IsStronglyConnected}\left(M\right)$

$\mathrm{false}$

$\mathrm{StronglyConnectedComponents}\left(M\right)$

$\left[\left[3\right]\,\left[1\,2\right]\,\left[4\,5\right]\right]$

$\mathrm{AdjacencyMatrix}\left(M\right)$

$\left[\begin{array}{rrrrr}0& 1& 1& 0& 0\\ 1& 0& 0& 1& 1\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1\\ 0& 0& 0& 1& 0\end{array}\right]$

$\mathrm{AllPairsDistance}\left(M\right)$

$\left[\begin{array}{ccccc}0& 1& 1& 2& 2\\ 1& 0& 2& 1& 1\\ \mathrm{\∞}& \mathrm{\∞}& 0& \mathrm{\∞}& \mathrm{\∞}\\ \mathrm{\∞}& \mathrm{\∞}& \mathrm{\∞}& 0& 1\\ \mathrm{\∞}& \mathrm{\∞}& \mathrm{\∞}& 1& 0\end{array}\right]$