How did you think of defining the dots as {i,i+1} etc.

Trial and error! The picture was symmetrical enough that it seemed likely that some systematic assignment of sets to the vertices would work, and {i,i+1} for five of the 2-element sets was a simple thing to start with. From there, I just played around with similar expressions for the other dots, until I found ones that met all the requirements.

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Why are we calling the laplacian of the graph?

When our graph is a n-dimensional lattice, my formula for gives us the usual discrete approximation to the Laplacian on . This is how you numerically compute a Laplacian using a computer. I explained this in a comment on my article about graph Laplacians.

In QM, has a laplacian term and a potential term. Are we considering the in

to be the potential term ?

No, all this is the Laplacian term. I’m considering a free particle, with no potential. But we can add a potential term if we want, too:

for some real-valued function defined on the vertices of the graph.

I did not understand how we are taking the same for the corresponding SM and QM problems?

It’s a generalization of how we use the Laplacian both in Schrödinger’s equation

(for a free particle in quantum mechanics) and the heat equation

(for a free particle in stochastic mechanics). I explained this at the start of this post.

Mathematically, the only difference between Schrödinger’s equation and the heat equation is the factor of But one describes how amplitudes change with time, while the other describes how probabilities change with time! This is well-known, but it’s a fascinating and mysterious thing. This whole series of notes is about taking techniques from quantum mechanics and applying them to stochastic mechanics, so it’s good to study their ‘overlap’.

In Part 16 I’ll explain that there’s a large class of operators such that is unitary and is stochastic. The most important example is when is the Laplacian. Graph Laplacians are also examples.

]]>In QM, H has a laplacian term and a potential term. Are we considering the in

to be the potential term?

I did not understand how we are taking the same for the corresponding SM and QM problems.

]]>… except I often haven’t transferred the *most recent* part to this site.

I’m glad you want to read it all!

]]>I would like to read the Network Theory series from the beginning, but my searches only return installments going back to part 4. Can you help?

Thank You,

Dave DeBenedetto