  For a $\Delta$-regular connected graph ${\sf H}$ the problem of determining
the upper tail large deviation for the number of copies of ${\sf H}$ in
$\mathbb{G}(n,p)$, an Erd\H{o}s-R\'{e}nyi graph on $n$ vertices with edge
probability $p$, has generated significant interests. For $p\ll 1$ and
$np^{\Delta/2} \gg (\log n)^{1/(v_{\sf H}-2)}$, where $v_{\sf H}$ is the number
of vertices in ${\sf H}$, the upper tail large deviation event is believed to
occur due to the presence of localized structures. In this regime the large
deviation event that the number of copies of ${\sf H}$ in $\mathbb{G}(n,p)$
exceeds its expectation by a constant factor is predicted to hold at a speed
$n^2 p^{\Delta} \log (1/p)$ and the rate function is conjectured to be given by
the solution of a mean-field variational problem. After a series of
developments in recent years, covering progressively broader ranges of $p$, the
upper tail large deviations for cliques of fixed size was proved by Harel,
Mousset, and Samotij \cite{hms} in the entire localized regime. This paper
establishes the conjecture for all connected regular graphs in the whole
localized regime.
