Large deviation principle for the norm of the Laplacian matrix of inhomogeneous Erdős-Rényi random graphs

We consider an inhomogeneous Erdős-Rényi random graph $G_N$ with vertex set $[N]=\{1,\dots ,N\}$ for which the pair of vertices $i,j\in [N]$, $i\mathrm{\ne }j$, is connected by an edge with probability $r_N(\frac{i}{N},\frac{j}{N})$, independently of other pairs of vertices. Here, $r_N\text{\hspace{0.17em}⁣}:[0,1{]}^2\to (0,1)$ is a symmetric function that plays the role of a reference graphon. Let ${\lambda }_N$ be the maximal eigenvalue of the Laplacian matrix of $G_N$. We show that if ${\lim }_{N\to \mathrm{\infty }}\mathrm{\parallel }{r}_N-r{\mathrm{\parallel }}_{\mathrm{\infty }}=0$ for some limiting graphon $r\text{\hspace{0.17em}⁣}:[0,1{]}^2\to (0,1)$, then ${\lambda }_N\mathrm{/}N$ satisfies a downward LDP with rate $\left(\genfrac{}{}{0px}{}{N}{2}\right)$ and an upward LDP with rate $N$. We identify the associated rate functions ${\psi }_r$ and ${\widehat{\psi }}_r$, and derive their basic properties.