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I'm having difficulty understanding a fact stated in a research paper I'm reading. Namely, let $T$ be a tree with all nodes of degree $4$ (ie, the root has $4$ daughter nodes and all other nodes have $3$ daughter nodes). The paper states without explanation that the spectrum of the Laplacian on $T$ is in $[4-2\sqrt{3},\infty)$.

How does one derive this result? I've searched around but haven't found many resources addressing this particular topic.

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  • $\begingroup$ Is this a metric graph, or are you considering the combinatorial Laplacian? Please state your definitions. $\endgroup$
    – user1688
    Jan 28, 2016 at 5:37
  • $\begingroup$ It would also be useful to give a precise reference to the paper you're reading. $\endgroup$
    – j.c.
    Jan 28, 2016 at 6:39

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A very lucid explanation is given in Luca Trevisan's blog.

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  • $\begingroup$ It looks like that page computes the spectral norm of the adjacency operator, from which I guess we can deduce the bottom of the spectrum of the Laplacian. But one would still have to prove that every larger number is also in the spectrum, right? That part doesn't seem to be done there. $\endgroup$ Jan 28, 2016 at 16:16
  • $\begingroup$ @NateEldredge If you read the OP's question, that is NOT what he is asking (although it is true that there is continuous spectrum in the interval - I think this is not so difficult to check by looking at eigenfunctions decaying exponentially radially from the root). And if by "deduce", you mean "subtract from $4$" yes, correct. $\endgroup$
    – Igor Rivin
    Jan 28, 2016 at 17:12
  • $\begingroup$ Oh, I missed the word "in". I thought OP was asking to show that the spectrum is $[4-2\sqrt{3}, \infty)$. $\endgroup$ Jan 28, 2016 at 17:13

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