Just run the next cell.  The graphics will show up at the bottom.

uPGsm2 = {-2.7884,-2.4752,2.4752,2.7884,0,2.4752,0,0,-2.4752,0};
uDGsm2 = {-2.7884,2.7884,-2.4752,0,2.4752,2.4752,-2.4752,0,0,0,2.4752,-2.4752,0,0,0,-2.7884,2.7884,-2.4752,0,2.4752};
uPMsm2={-2.7884,-2.4752,2.4752,2.7884,-2,2.4752,2,2,-2.4752,-2};
uDMsm2 = {-2.7884,2.7884,-2.4752,2,2.4752,2.4752,-2.4752,-2,2,-2,2.4752,-2.4752,-2,2,-2,-2.7884,2.7884,-2.4752,2,2.4752};
uPD1sm2 ={-2.7828,-2.4626,2.4877,2.7932,-2.186,2.3543,2.0137,2.0113,-2.5815,-0.509};
uDD1sm2 ={-2.7828,2.7932,-2.5815,2.0113,2.4877,2.3543,-2.4626,-0.509,2.0137,-2.186,2.3543,-2.4626,-0.509,2.0137,-2.186,-2.7828,2.7932,-2.5815,2.0113,2.4877};uPD2sm2 ={-2.7884,-2.3369,2.4754,2.7874,-2.2003,2.4749,0.4908,2.198,-2.5949,-0.4872};
uDD2sm2 = {-2.7884,2.7874,-2.5949,2.198,2.4754,2.4749,-2.3369,-0.4872,0.4908,-2.2003,2.4749,-2.3369,-0.4872,0.4908,-2.2003,-2.7884,2.7874,-2.5949,2.198,2.4754};
PetersenCoo = {{0,0,1}, {1,1,-1}, {-1,1,1}, {0,0,-1}, {0,1,0},{1, -1, 1},{1,0,0},{-1,0,0}, {-1,-1,-1},{0, -1, 0}};

dodecaCoo = {{-0.726542528001432`,-0.23606797750932015`,-1.`},{0,-0.763932022508271`,-1.`},{0.7265425280107184`,-0.23606797750932004`,-1.`},{0.44902797659219124`,0.6180339887379862`,-1.`},{-0.4490279765678626`,0.6180339887379861`,-1`},{-1.175570504587879`,-0.3819660112514467`,-0.23606797749979003`},{0,-1.236067977498753`,-0.2360679774997902`},{1.1755705045843252`,-0.3819660112514466`,-0.23606797749978978`},{0.7265425280164681`,1.`,-0.23606797749978964`},{-0.7265425279956824`,1.`,-0.2360679774997896`},{1.1755705045878793`,0.38196601125144664`,0.23606797749978994`},{0,1.236067977498753`,0.2360679774997902`},{-1.1755705045843257`,0.3819660112514466`,0.23606797749978978`},{-0.7265425280164681`,-1.`,0.23606797749978944`},{0.7265425279956825`,-1.`,0.23606797749978956`},{0.726542528001432`,0.2360679775093202`,1.`},{0,0.7639320225082713`,1.`},{-0.7265425280107184`,0.23606797750932004`,1.`},{-0.44902797659219124`,-0.6180339887379862`,1.`},{0.44902797656786253`,-0.6180339887379858`,1.`}};
makeA[edgeList_]:= Module[{A,e, i, j,n},
n = Max@Flatten[edgeList];
A = Table[0, {i,n}, {j,n}];
Do[ {i, j} = edgeList[[e]];
A[[i,j]] = 1;
A[[j,i]] = 1, 
{e, Length[edgeList]}];
A]
PetersenEdgeList = {{1,3}, {1,4}, {1,6}, {2,4}, {2,5}, {2,7}, {3,5}, {3,8}, {4,9}, {5,10},{6,7},{6,10}, {7,8}, {8,9}, {9,10}};
PetersenA = makeA[PetersenEdgeList];
dodecaEdgeList = {{1,2},{2,3},{3,4},{4,5},{5,1},{1,6},{2,7},{3,8},{4,9},{5,10},{11,16},{12,17},{13,18},{14,19},{15,20},{6,13},{6,14},{7,14},{7,15},{8,11},{8,15},{9,11},{9,12},{10,12},{10,13},{16,17},{17,18},{18,19},{19,20},{20,16}};
dodecaA = makeA[dodecaEdgeList];
view3D[x_, A_, u_,ratio_] := Module[{graphicsList = {}, n = Length[x], m = Length[x[[1]] ], maxDist,xt, x3D,rad, L, vi, vj, uMax = Max@Abs[u]},
xt = Transpose[x];
Which[m == 1, x3D = Transpose@Join[xt, Table[0, {2}, {n}]],
m == 2, x3D = Transpose@AppendTo[xt, Table[0, {n}]],
m == 3, x3D = x,
m > 3, x3D = Transpose@Take[xt,3]];
maxDist = Max@Abs[x3D];
rad = ratio*maxDist;
L = maxDist + rad;
Do[
vi = x3D[[i]];
radius = Abs[rad*Sqrt[u[[i]]/uMax]];
If[u[[i]] > 0, color=Red, color =Cyan];
If[radius < 0.1*rad, radius = 0.25*rad; color = Black];
AppendTo[graphicsList, {color,Sphere[vi, radius]}];
Do[
vj = x3D[[j]];
If[A[[i]][[j]]==1,AppendTo[graphicsList, {LightGray,Cylinder[{vi, vj}, rad/10]}]]
,{j,i+1, n}]
, {i, n}];
Graphics3D[graphicsList, Boxed -> True, SphericalRegion -> True, PlotRange -> {{-L,L}, {-L,L}, {-L,L}}]];
Manipulate[If[graph == "Petersen", 
{coo = PetersenCoo;
A = PetersenA;
ratio = 0.2;
Which[branch == "Grandmother", u = uPGsm2,
branch == "Mother", u = uPMsm2,
branch == "Daughter1", u = uPD1sm2,
branch == "Daughter2", u = uPD2sm2]
}];
If[graph == "Dodecahedron", 
{coo = dodecaCoo;
A = dodecaA;
ratio = 0.15;
Which[branch == "Grandmother", u = uDGsm2,
branch == "Mother", u = uDMsm2,
branch == "Daughter1", u = uDD1sm2,
branch == "Daughter2", u = uDD2sm2]
}];
view3D[coo, A,u, ratio], {graph, {"Petersen", "Dodecahedron"}}, {branch, {"Grandmother","Mother", "Daughter1", "Daughter2"}}]