Finite projective planes meet spectral gaps

We show that for any connected graph $G$ with maximum degree $d\ge3$, the spectral gap from $0$ with respect to the adjacency matrix is at most $\sqrt{d-1}$, with equality if and only if $G$ is the incidence graph of a finite projective plane of order $d-1$; and for other cases, the bound $\sqrt{d-1}$ is improved to $\sqrt{d-2}$. This is a spectral gap version of a result by Mohar and Tayfeh-Rezaie. Moreover, for $d$-regular graphs with girth at least 7, the bound $\sqrt{d-2}$ is further improved to $\sqrt{d-c(d)}$ where $c(d)\ge 2$ and $\lim\limits_{d\to\infty}c(d)/d=(\sqrt{5}-1)/2$. A similar yet more subtle phenomenon involving the normalized Laplacian is also investigated, where we work on graphs of degrees $\ge d$ rather than $\le d$. We prove that for any graph $G$ with \emph{minimum} degree $d\ge 3$, the spectral gap from the value 1 with respect to the normalized Laplacian is at most $\sqrt{d-1}/d$, with equality if and only if $G$ is the incidence graph of a finite projective plane of order $d-1$. As an application, we provide a new sharp bound for the convergence rate of some eigenvalues of the Laplacian on the weighted neighborhood graphs introduced by Bauer and Jost.