I was curious about the limit of maximum eigenvalue of the laplacian for some regular infinite graph, and here the graph is a 4-Cayley Tree $T_\infty$.
The 4-Cayley Tree $T_k$ is a depth k complete tree that each non-leaf vertex has 4 branches, you can find reference here, and the figure of a depth 3 4-cayley tree $T_3$ is illustrated below. The infinity depth 4-cayley tree $T_\infty$ is a Bethe lattice
I used python package networkx and numpy to calculate the first several cases of the maximum eigenvalue of the laplacian of $T_k$, and here are the results
| k | nodes | maximum eigenvalue |
|---|---|---|
| 1 | 5 | 5.0 |
| 2 | 17 | 6.302775637731994 |
| 3 | 53 | 6.773387411650647 |
| 4 | 161 | 6.999999999999993 |
| 5 | 485 | 7.128330434356192 |
| 6 | 1457 | 7.2088130934416705 |
| 7 | 4373 | 7.262941326929419 |
| 8 | 13121 | 7.301244084793453 |
It seems that the maximum eigenvalue is approaching to $e^2$ which is 7.3890560989306495...
So, in all, what is the maximum eigenvalue of $T_\infty$? and any related references?