Delio's Second Example
L = [3 4 sqrt(10) pi];
numEigs=6;
The basic graph
Calculate and display the first few eigenvalues and eigenfunctions;
Phi1= quantumGraphFromTemplate('balalaika',Lvec=L,discretization='Chebyshev');
Phi1.plot('layout');
[V1,d1]=Phi1.eigs(numEigs);
for k=1:numEigs
figure
Phi1.plot(V1(:,k));
title(sprintf('$\\omega_{%i} = %0.3f$',k,d1(k)))
end
The graph with all vertices merged to make a clover
This should have larger (more negative) eigenvalues
Phi2= quantumGraphFromTemplate('clover',nLeaves=4,Lvec=L,discretization='Chebyshev');
Phi2.plot('layout');
[V2,d2]=Phi2.eigs(numEigs);
for k=1:numEigs
figure
Phi2.plot(V2(:,k));
title(sprintf('$\\omega_{%i} = %0.3f$',k,d2(k)))
end
A new clover with only the two longest edges of the previous graph
This should have even larger negative eigenvalues
L=sort(L);
L = L(3:4);
Phi3= quantumGraphFromTemplate('clover',nLeaves=2,Lvec=L,discretization='Chebyshev');
Phi3.plot('layout');
[V3,d3]=Phi3.eigs(numEigs);
for k=1:numEigs
figure
Phi3.plot(V3(:,k));
title(sprintf('Eigenfunction %i, $\\omega = %0.3f$',k,d3(k)))
end
Comparing the eigenvalues
nn = 0:numEigs-1;
plot(nn,d1,'-o',nn,d2,'--o',nn,d3,'-.o')%,nn,w,'x')
xlabel('$k$')
ylabel('$\omega_k$')
legend('Graph 1','Graph 2','Graph 3','Location','southwest')
