# The spectrum of the infinite tree

The spectral norm of the infinite ${d}$-regular tree is ${2 \sqrt {d-1}}$. We will see what this means and how to prove it.

When talking about the expansion of random graphs, abobut the construction of Ramanujan expanders, as well as about sparsifiers, community detection, and several other problems, the number ${2 \sqrt{d-1}}$ comes up often, where ${d}$ is the degree of the graph, for reasons that tend to be related to properties of the infinite ${d}$-regular tree.