$\mathrm{vertices}≔\left[''Alice''\, ''Bob''\, ''Andrew''\, ''Ned''\, ''Fred''\, ''Gerald''\, ''Sarah''\, ''Herbet''\, ''Cormac''\, ''Hopkins''\, ''Maggie''\right]\:$

$\mathrm{edges\_weights}≔ \left\{\left[\left\{''Hopkins''\, ''Fred''\right\}\, 9\right]\, \left[\left\{''Hopkins''\, ''Andrew''\right\}\, 18\right]\, \left[\left\{''Bob''\, ''Fred''\right\}\, 27\right]\, \left[\left\{''Bob''\, ''Andrew''\right\}\, 14\right]\, \left[\left\{''Bob''\, ''Alice''\right\}\, 25\right]\, \left[\left\{''Bob''\, ''Herbet''\right\}\, 33\right]\, \left[\left\{''Bob''\, ''Ned''\right\}\, 3\right]\, \left[\left\{''Bob''\, ''Cormac''\right\}\, 5\right]\, \left[\left\{''Maggie''\, ''Andrew''\right\}\, 51\right]\, \left[\left\{''Fred''\, ''Alice''\right\}\, 12\right]\, \left[\left\{''Fred''\, ''Herbet''\right\}\, 38\right]\, \left[\left\{''Fred''\, ''Ned''\right\}\, 5\right]\, \left[\left\{''Sarah''\, ''Alice''\right\}\, 6\right]\, \left[\left\{''Sarah''\, ''Herbet''\right\}\, 3\right]\, \left[\left\{''Gerald''\, ''Andrew''\right\}\, 4\right]\, \left[\left\{''Gerald''\, ''Herbet''\right\}\, 3\right]\, \left[\left\{''Andrew''\, ''Alice''\right\}\, 4\right]\, \left[\left\{''Andrew''\, ''Herbet''\right\}\, 44\right]\, \left[\left\{''Andrew''\, ''Ned''\right\}\, 9\right]\, \left[\left\{''Alice''\, ''Herbet''\right\}\, 112\right]\, \left[\left\{''Alice''\, ''Ned''\right\}\, 5\right]\, \left[\left\{''Herbet''\, ''Ned''\right\}\, 11\right]\right\}\:$

$\mathrm{Gs}≔\mathrm{Graph}\left(\mathrm{vertices}\,\mathrm{edges\_weights}\right)$

$\mathrm{Gs}≔\mathrm{Graph\; 1:\; an\; undirected\; weighted\; graph\; with\; 11\; vertices\; and\; 22\; edge(s)}$

$\mathrm{DrawGraph}\left(\mathrm{Gs}\, \mathrm{layout}\=\mathrm{spring}\,\mathrm{stylesheet}\=\left[\mathrm{vertexcolor} \= ''Niagara\; DeepBlue'' \,\mathrm{vertexfontcolor} \= \mathrm{white} \,\mathrm{vertexborder} \= \mathrm{false} \,\mathrm{vertexshape} \= ''stadium'' \,\mathrm{vertexfont} \= \left[''Arial''\, 10\right] \,\mathrm{vertexpadding} \= 8 \,\mathrm{edgecolor} \= ''Niagara\; Burgundy''\right]\,\mathrm{title}\=''Social\; Media\; Graph''\,\mathrm{titlefont}\=\left[\mathrm{Arial}\,16\right]\,\mathrm{size}\=\left[600\,600\right]\right)$

$\mathrm{centrality}≔\mathrm{EigenvectorCentrality}\left(\mathrm{Gs}\right)$

$\mathrm{centrality}≔\left[0.237978443268371\,0.131195758579289\,0.116408170276784\,0.0410886760673835\,0.113645015600347\,0.00850460849139520\,0.0148924786849715\,0.271820184196355\,0.00435475160238433\,0.0207000263726866\,0.0394118868600335\right]$

$\mathrm{ind}≔\mathrm{sort}\left(\mathrm{centrality}\,\mathrm{`>`}\,\mathrm{output}\=\mathrm{permutation}\right)\:$

$\mathrm{Statistics}:-\mathrm{BarChart}\left(\left[\mathrm{seq}\left(\mathrm{vertices}\left[i\right]=\mathrm{centrality}\left[i\right]\,i \mathbf{in} \mathrm{ind}\right)\right]\,\mathrm{title}=''Most\; Influential\; Social\; Media\; Users''\,\mathrm{titlefont}=\left[\mathrm{Arial}\,16\right]\,\mathrm{font}=\left[\mathrm{Arial}\,10\right]\,\mathrm{axesfont}=\left[\mathrm{Arial}\,10\right]\,\mathrm{labels}=\left[''Centrality''\,''''\right]\,\mathrm{labelfont}=\left[\mathrm{Arial}\,14\right]\,\mathrm{axes}=\mathrm{frame}\,\mathrm{color}=\mathrm{ColorTools}:-\mathrm{Color}\left(\frac{\left[0\, 79\, 128\right]}{255}\right)\,\mathrm{size}=\left[600\,400\right]\,\mathrm{style}=\mathrm{surface}\right)$

$S ≔ \mathrm{SpecialGraphs}:-\mathrm{SoccerBallGraph}\left(\right)\;$

$S≔\mathrm{Graph\; 2:\; an\; undirected\; unweighted\; graph\; with\; 60\; vertices\; and\; 90\; edge(s)}$

$\mathrm{DrawGraph}\left(S\, \mathrm{stylesheet}\=\left[\mathrm{vertexcolor}\=''Yellow''\, \mathrm{vertexfontcolor}\=''Red''\right]\, \mathrm{dimension}\=3\right)$

$P ≔ \mathrm{SpecialGraphs}:-\mathrm{PetersenGraph}\left(\right)\;$

$P≔\mathrm{Graph\; 4:\; an\; undirected\; unweighted\; graph\; with\; 10\; vertices\; and\; 15\; edge(s)}$

$\mathrm{DrawGraph}\left(P\, \mathrm{layout}\=\mathrm{spring}\, \mathrm{animate}\=25\, \mathrm{stylesheet}\=\left[\mathrm{vertexcolor}\=''SeaGreen''\, \mathrm{vertexborder}\=''DarkSeaGreen''\right]\right)\;$

$\mathrm{Gd} ≔ \mathrm{Graph}\left(\mathrm{Matrix}\left(\left[\left[0\,2\,9\right]\,\left[4\,0\,1\right]\,\left[3\,0\,0\right]\right]\right)\right)$

$\mathrm{Gd}≔\mathrm{Graph\; 5:\; a\; directed\; weighted\; graph\; with\; 3\; vertices\; and\; 5\; arc(s)}$

$$\begin{gathered} \mathrm{StyleEdge}\left(\mathrm{Gd}\, \left[2\,3\right]\, \mathrm{fontcolor}\=''\_match''\, \mathrm{color}\=''\_blend''\, \mathrm{arrowshape}\=''line''\, \mathrm{arrowpos}\=0\right)\; \\ \mathrm{StyleVertex}\left(\mathrm{Gd}\, \left[2\,3\right]\, \mathrm{color}\=''Red''\,\mathrm{shape}\=''square''\right)\; \end{gathered}$$

$\mathrm{StyleVertex}\left(\mathrm{Gd}\, 2\, \mathrm{color}\=''Blue''\right)\;$

$\mathrm{DrawGraph}\left(\mathrm{Gd}\right)\;$

eigenvector centrality of a vertex, we linearly transition between two colors and two font sizes.

weight of an edge, we linearly transition between two colors and thicknesses.

$$\begin{gathered} \mathrm{c1}≔ ''DarkSeaGreen''\: \\ \mathrm{c2}≔''SeaGreen''\: \\ \mathrm{StyleVerticesByProperty}\left(\mathrm{Gs}\,\mathrm{EigenvectorCentrality}\, \mathrm{fontsizescheme}\=\left[10\,16\right]\,\mathrm{colorscheme}\=\left[\mathrm{c1}\, \mathrm{c2}\right]\right)\: \\ \mathrm{StyleEdgesByProperty}\left(\mathrm{Gs}\,\mathrm{WeightMatrix}\,\mathrm{colorscheme}\=\left[\mathrm{c1}\, \mathrm{c2}\right]\,\mathrm{thicknessscheme}\=\left[1\,6\right]\right)\: \end{gathered}$$

$\mathrm{DrawGraph}\left(\mathrm{Gs}\,\mathrm{style}\=\mathrm{spring}\,\mathrm{stylesheet}\=\left[\mathrm{vertexfontcolor}=\mathrm{white}\,\mathrm{vertexborder}=\mathrm{false}\,\mathrm{vertexpadding}\=4\right]\,\mathrm{title}=''Social\; Media\; Graph''\,\mathrm{titlefont}=\left[\mathrm{Arial}\,16\right]\,\mathrm{size}=\left[600\,600\right]\,\mathrm{showweights}=\mathrm{false}\right)$

for vertices, we assign specific colors and font sizes to ranges of the eigenvector centrality.

for edges, we assign specific colors and thicknesses to ranges of the weight.

$\mathrm{c1}≔''\#67809F''\:$$\mathrm{c2}≔''\#004F79''\:$$\mathrm{c3}≔''\#96281B''\:$$\mathrm{StyleVerticesByProperty}\left(\mathrm{Gs}\,\mathrm{EigenvectorCentrality}\,\mathrm{colorscheme}\=\left[''valuesplit''\,\left[..0.1=\mathrm{c1}\,0.1..0.25=\mathrm{c2}\,\mathrm{c3}\right]\right]\,\mathrm{fontsizescheme}\=\left[''valuesplit''\,\left[..0.2\=10\,18\right]\right]\right)\;$$\mathrm{StyleEdgesByProperty}\left(\mathrm{Gs}\,\mathrm{WeightMatrix}\,\mathrm{colorscheme}\=\left[''valuesplit''\,\left[..20=\mathrm{c1}\,\mathrm{c2}\right]\right]\,\mathrm{thicknessscheme}\=\left[''valuesplit''\,\left[..10=1\,10..20\=2\,4\right]\right]\right)\:$

$\mathrm{DrawGraph}\left(\mathrm{Gs}\,\mathrm{style}\=\mathrm{spring}\,\mathrm{stylesheet}\=\left[\mathrm{vertexfontcolor}=\mathrm{white}\,\mathrm{vertexborder}=\mathrm{false}\,\mathrm{vertexpadding}\=4\right]\,\mathrm{title}=''Social\; Media\; Graph''\,\mathrm{titlefont}=\left[\mathrm{Arial}\,16\right]\,\mathrm{size}=\left[600\,600\right]\,\mathrm{showweights}=\mathrm{false}\right)$

$\mathrm{Gp} ≔ \mathrm{GraphTheory}:-\mathrm{SpecialGraphs}:-\mathrm{PetersenGraph}\left(\right)\;$

$\mathrm{Gp}≔\mathrm{Graph\; 7:\; an\; undirected\; unweighted\; graph\; with\; 10\; vertices\; and\; 15\; edge(s)}$

$\mathrm{DrawGraph}\left( \mathrm{Gp}\, \mathrm{layout}\=\mathrm{spring}\, \mathrm{layoutoptions}\=\left[\mathrm{initial}\=\mathrm{circle}\, \mathrm{constantonly}\=\mathrm{true}\right] \right)\;$

$\mathrm{Gr} ≔ \mathrm{RandomGraphs}:-\mathrm{RandomGraph}\left(10\, 25\right)\;$

$\mathrm{Gr}≔\mathrm{Graph\; 3:\; an\; undirected\; unweighted\; graph\; with\; 10\; vertices\; and\; 25\; edge(s)}$

$\mathrm{DrawGraph}\left(\mathrm{Gr} \, \mathrm{layout}\= \mathrm{interactive}\, \mathrm{layoutoptions}\=\left[\mathrm{initial}\=\mathrm{spring}\right] \right)$

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

$\mathrm{G1}≔\mathrm{RandomGraphs}:-\mathrm{RandomGraph}\left(100\, 250\right)$

$\mathrm{G1}≔\mathrm{Graph\; 8:\; an\; undirected\; unweighted\; graph\; with\; 100\; vertices\; and\; 250\; edge(s)}$

$\mathrm{DrawGraph}\left(\mathrm{G1}\,\mathrm{layout}\=\mathrm{spectral}\right)$

$\mathrm{DrawGraph}\left(\mathrm{G1}\,\mathrm{layout}\=\mathrm{spectral}\,\mathrm{dimension}\=3\right)$

$G ≔\mathrm{SpecialGraphs}:-\mathrm{DeBruijnGraph}\left(5\,3\right)$

$G≔\mathrm{Graph\; 11:\; a\; directed\; unweighted\; graph\; with\; 125\; vertices,\; 620\; arc(s),\; and\; 5\; self-loop(s)}$

HasSelfLoops returns true if a graph has an edge or arc to itself.

NumberOfSelfLoops returns the number of self-loops in a graph.

SelfLoops returns the set of self-loops in a graph.

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

$\mathrm{true}$

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

$5$

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

$\left\{1\,32\,63\,94\,125\right\}$

$\mathrm{G1} ≔ \mathrm{Graph}\left(\left\{\left[1\,1\right]\,\left[1\,2\right]\,\left[1\,3\right]\,\left[1\,4\right]\,\left[3\,4\right]\,\left[2\,3\right]\right\}\right)$

$\mathrm{G1}≔\mathrm{Graph\; 8:\; a\; directed\; unweighted\; graph\; with\; 4\; vertices,\; 5\; arc(s),\; and\; 1\; self-loop(s)}$

$\mathrm{DrawGraph}\left(\mathrm{G1}\right)$

$\mathrm{NumberOfSelfLoops}\left(\mathrm{G1}\right)$

$1$

$\mathrm{Edges}\left(\mathrm{G1}\right)$

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

$\mathrm{Edges}\left(\mathrm{G1}\,\mathrm{selfloops}\=\mathrm{false}\right)$

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

$\mathrm{InDegree}\left(\mathrm{G1}\,1\right)$

$1$

$\mathrm{OutDegree}\left(\mathrm{G1}\,1\right)$

$4$

$\mathrm{G2}≔ \mathrm{UnderlyingGraph}\left(\mathrm{G1}\,\mathrm{selfloops}\right)$

$\mathrm{G2}≔\mathrm{Graph\; 9:\; an\; undirected\; unweighted\; graph\; with\; 4\; vertices,\; 5\; edge(s),\; and\; 1\; self-loop(s)}$

$\mathrm{DrawGraph}\left(\mathrm{G2}\right)$

$\mathrm{DegreeSequence}\left(\mathrm{G2}\right)$

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

$\mathrm{Girth}\left(\mathrm{G2}\right)$

$1$

$\mathrm{ChromaticPolynomial}\left(\mathrm{G2}\,x\right)$

$0$

$\mathrm{G3}≔ \mathrm{UnderlyingGraph}\left(\mathrm{G1}\right)$

$\mathrm{G3}≔\mathrm{Graph\; 10:\; an\; undirected\; unweighted\; graph\; with\; 4\; vertices\; and\; 5\; edge(s)}$

$\mathrm{NumberOfSelfLoops}\left(\mathrm{G3}\right)$

$0$

$\mathrm{DegreeSequence}\left(\mathrm{G3}\right)$

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

$\mathrm{Girth}\left(\mathrm{G3}\right)$

$3$

$\mathrm{ChromaticPolynomial}\left(\mathrm{G3}\,x\right)$

$x\left(x-1\right){\left(x-2\right)}^2$

$\mathrm{G5}≔\mathrm{RandomGraphs}:-\mathrm{RandomGeometricGraph}\left(10\,2\,\mathrm{distribution}\=\mathrm{Weibull}\left(3\,2\right)\,\mathrm{weighted}\right)$

$\mathrm{DrawGraph}\left(\mathrm{G5}\right)$$$

$\mathrm{with}\left(\mathrm{GeometricGraphs}\right)\;$

$\left[\mathrm{DelaunayGraph}\,\mathrm{EuclideanMinimumSpanningTree}\,\mathrm{FarthestNeighborGraph}\,\mathrm{GabrielGraph}\,\mathrm{GeometricMinimumSpanningTree}\,\mathrm{NearestNeighborGraph}\,\mathrm{RelativeNeighborhoodGraph}\,\mathrm{SphereOfInfluenceGraph}\,\mathrm{UnitDiskGraph}\,\mathrm{UrquhartGraph}\,\mathrm{VisibilityGraph}\right]$

$\mathrm{points}≔\mathrm{Matrix}\left(50\,2\,\mathrm{rand}\left(0..100.\right)\right)\:$

The Euclidean minimum spanning tree is a subgraph of the relative neighborhood graph,

The relative neighborhood graph is a subgraph of the Urquhart graph,

The Urquhart graph is a subgraph of the Gabriel graph.

The Gabriel graph is a subgraph of the Delaunay graph.

$\mathrm{DrawGraph}\left(\mathrm{UnitDiskGraph}\left(\mathrm{points}\,15\right)\right)$

$\mathrm{DrawGraph}\left(\mathrm{SphereOfInfluenceGraph}\left(\mathrm{points}\right)\right)$

$$\begin{gathered} \mathrm{G3}≔\mathrm{Graph}\left(5\,\left\{\left\{1\,2\right\}\,\left\{1\,3\right\}\,\left\{2\,3\right\}\,\left\{2\,4\right\}\,\left\{3\,4\right\}\,\left\{4\,5\right\}\right\}\right)\; \\ \mathrm{DrawGraph}\left(\mathrm{G3}\right)\; \end{gathered}$$

$\mathrm{G3}≔\mathrm{Graph\; 12:\; an\; undirected\; unweighted\; graph\; with\; 5\; vertices\; and\; 6\; edge(s)}$

$\mathrm{result}\, \mathrm{split} ≔ \mathrm{IsSplitGraph}\left(\mathrm{G3}\,\mathrm{decomposition}\right)$

$\mathrm{result},\mathrm{split}≔\mathrm{true},\left[\left[2\,3\,4\right]\,\left[1\,5\right]\right]$

$\mathrm{HighlightSubgraph}\left(\mathrm{G3}\,\mathrm{InducedSubgraph}\left(\mathrm{G3}\,\mathrm{split}\left[1\right]\right)\right)$

$\mathrm{DrawGraph}\left(\mathrm{G3}\right)$

$C≔\mathrm{CycleGraph}\left(6\right)\:$$H≔\mathrm{ContractSubgraph}\left(C\,\left[1\,2\,6\right]\right)$

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

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

$\mathrm{with}\left(\mathrm{SpecialGraphs}\right)\:$