Mugnolo's First Example

A balalaika-shaped graph

L = [3 4 sqrt(10) pi];
numEigs=6;

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

Note all the first 4 eigenvalues and eigenvectors are unchanged but numbers 5 and 6 feature split edges and smaller frequencies

Phi2= quantumGraphFromTemplate('balalaikaDoubled',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

All the frequencies are smaller

Phi3= quantumGraphFromTemplate('balalaikaBroken',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

Ltotal = 2*sum(L);

nn = 0:numEigs-1;

w= -ceil(nn/2).^2*4*pi^2/Ltotal^2;

plot(nn,d1,'-o',nn,d2,'--o',nn,d3,'-.o',nn,w,'x')

exactString=('$-\frac{4 \lceil k\rceil^2\pi^2}{L^2}$');

xlabel('$k$')

ylabel('$\omega_k$')

legend('Graph 1','Graph 2','Graph 3',exactString,'Location','southwest')