Ilić and Ilić's minimum Laplacian-coefficient conjecture for unicyclic graphs

Let $\mathcal{U}_{n,l}$ be the set of $n$-vertex unicyclic graphs with $l$ leaves, and let $U_{n,l}^{3,0}$ be the balanced starlike unicyclic graph of order $n$, with $l$ leaves and girth $3$, obtained with attachment parameter $p=0$. For graphs on $n$ vertices, write $G\preceq H$ when every Laplacian coefficient of $G$ is at most the corresponding coefficient of $H$.

Ilić and Ilić's conjecture. Among all $n$-vertex unicyclic graphs, the graph $U_{n,l}^{3,0}$ has the minimum Laplacian coefficients $c_k$, $k=0,\dots,n$; equivalently, $U_{n,l}^{3,0}$ is the only minimal element in the poset $(\mathcal{U}_{n,l},\preceq)$.

The conjecture concerns the simultaneous minimization of all Laplacian coefficients within unicyclic graphs with a fixed number of leaves. It is explicitly stated later in the paper to be false, so the original conjecture is refuted.