I'm reading this paper and am slightly puzzled by the proof of Theorem 1.2 https://arxiv.org/abs/2007.12819 .
Theorem 1.2 states as follows:
If $G$ is a connected graph of maximum degree $\Delta$ on $n$ vertices with $\lambda_2(A_G)=\lambda_2$, then the number of eigenvalues in the range $$ [(1-\frac{log\,log_\Delta n}{log_\Delta n})\lambda_2,\lambda_2] $$ of the normalised adjacency matrix $A_G$ is $O(n\frac{\Delta^{7/5}}{log^{1/5}n})$ (hiding polylogarithmic factors).
In the proof of the above (see page 14 of the arXiv paper) the authors claim that it is sufficient to assume that the diameter of $G$ is at least $4$. Otherwise on would have that $\Delta\geq n^{1/4}$ and the Theorem's statement would be vacuous. Why is this so? are there any known results about the multiplicity of eigenvalues of a graph with bounded diameter?