  Consider the inhomogeneous Erd\H{o}s-R\'enyi random graph (ERRG) on $n$
vertices for which each pair $i,j\in\{1,\ldots,n\}$, $i\neq j$ is connected
independently by an edge with probability $r_n(\frac{i-1}{n},\frac{j-1}{n})$,
where $(r_n)_{n\in\mathbb{N}}$ is a sequence of graphons converging to a
reference graphon $r$. As a generalization of the celebrated LDP for ERRGs by
Chatterjee and Varadhan (2010), Dhara and Sen (2019) proved a large deviation
principle (LDP) for a sequence of such graphs under the assumption that $r$ is
bounded away from 0 and 1, and with a rate function in the form of a lower
semi-continuous envelope. We further extend the results by Dhara and Sen. We
relax the conditions on the reference graphon to $\log r,\log(1-r)\in
L^1([0,1]^2)$. We also show that, under this condition, their rate function
equals a different, more tractable rate function. We then apply these results
to the large deviation principle for the largest eigenvalue of inhomogeneous
ERRGs and weaken the conditions for part of the analysis of the rate function
by Chakrabarty, Hazra, Den Hollander and Sfragara (2020).
