Faber–Krahn conjecture for unicyclic degree sequences

Let $\pi=(d_0,d_1,\ldots,d_{k-1},1,\ldots,1)$ be a graphic unicyclic degree sequence with $2\le d_0\le d_1\le\ldots\le d_{k-1}$ and $d_k=\ldots=d_{n-1}=1$. Let $\mathcal{U}_{\pi}$ denote the class of unicyclic graphs with degree sequence $\pi$, let $U_{\pi,B}^*$ be the specified extremal graph, and let $\lambda^D(G)$ denote the first Dirichlet eigenvalue of $G$. Faber–Krahn conjecture. For every $G\in\mathcal{U}_{\pi}$,

$$\lambda^D(G)\ge\lambda^D(U_{\pi,B}^*)$$

with equality if and only if $G=U_{\pi,B}^*$. Equivalently, $U_{\pi,B}^*$ is the unique graph with the Faber–Krahn property in $\mathcal{U}_{\pi}$. The preceding theorem establishes this assertion when $d_0\ge3$; the conjecture concerns the remaining case $d_0=2$.