The following Maple code generating 5 nodes and 9 edges has been used in class for demonstration. You may use it as an archetype to produce your own code. You may also use Matlabs commands digraph and centrality or any other software packages that are at your disposal to answer the work.

restart;

with(GraphTheory);

G:=Graph([P1,P2,P3,P4,P5],{[P1,P2],[P2,P1],[P2,P3],[P2,P4],[P3,P4],[P3,P5],[P4,P5],[P5,P1],[P5,P4]});

DrawGraph(G);

AR:=Arrivals(G);

Authority:=map(numelems, AR);

A:= AdjacencyMatrix(G);

DP:=Departures(G); Hub:=map(numelems, DP);

n:=5;

Google:=A;

for i to n do

s:=numelems(DP[i]);

for j to n do

Google[i, j]:=A[i, j]/s;

end do;

end do;

Google;

with(LinearAlgebra);

B:=Transpose(Google);

Q:=Vector([1/5, 1/5, 1/5, 1/5, 1/5]);

for k to 10 do Q:=MatrixVectorMultiply(B, Q);

end do;

evalf(Q);

a) Generate a random directed graph G with 10 nodes. There are 90 possible ways to link the nodes (Why?). Your graph should contain at least 30 edges. Use any software at your disposal. Report your graph by a computer generated graph. Your drawing should identify the nodes names and the directed edges. Make sure that there is no isolated, nor dangling, nodes, i.e., every node must have at least one outgoing and one incoming edges.

b) Use a table to report the authority scores and the hub scores of each node.

c)Generate the associated adjacency matrix L