A stability version for a theorem of Erd\H{o}s on nonhamiltonian graphs

Let $n,d$ be integers with $1 \leq d \leq \left \lfloor \frac{n-1}{2} \right \rfloor$,andset$h(n,d):={n-d \choose 2} + d^2$and$e(n,d):= \max{h(n,d),h(n, \left \lfloor \frac{n-1}{2} \right \rfloor)}$\mathrm{.}Because$h(n,d)$isquadraticin$d$,thereexistsa$d_0(n)=(n/6)+O(1)$suchthat$e(n,1)> e(n, 2)> \dots >e(n,d_0)=e(n, d_0+1)=\dots = e(n,\left \lfloor \frac{n-1}{2} \right \rfloor)$\mathrm{.}AtheorembyErd\stackrel{˝}{o}sstatesthatfor$d\leq \left \lfloor \frac{n-1}{2} \right \rfloor$,any$n$-vertex nonhamiltonian graph $G$ with minimum degree $\delta (G)\ge d$ has at most $e(n,d)$ edges, and for $d>d_0(n)$ the unique sharpness example is simply the graph $K_n-E(K_{\lceil (n+1)\mathrm{/}2\rceil })$. Erd\H{o}s also presented a sharpness example $H_{n,d}$ for each $1\le d\le d_0(n)$. We show that if $d<d_0(n)$ and a $2$-connected, nonhamiltonian $n$-vertex graph $G$ with $\delta (G)\ge d$ has more than $e(n,d+1)$ edges, then $G$ is a subgraph of $H_{n,d}$. Note that $e(n,d)-e(n,d+1)=n-3d-2\ge n\mathrm{/}2$ whenever $d<d_0(n)-1$.