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Let $G$ be a graph of order $n$ and let $\mathcal{L}(G,\lambda)=\sum_{k=0}^n(-1)^{k}c_{k}(G)\lambda^{n-k}$ be the characteristic polynomial of itsLaplacian matrix. Motivated by Ili\'{c} and Ili\'{c}'s conjecture [A. Ili\'{c},M. Ili\'{c}, Laplacian coefficients of trees with given number of leaves orvertices of degree two, Linear Algebra and its Applications431(2009)2195-2202.] on all extremal graphs which minimize all the Laplaciancoefficients in the set $\mathcal{U}_{n,l}$ of all $n$-vertex unicyclic graphswith the number of leaves $l$, we investigate properties of the minimalelements in the partial set $(\mathcal{U}_{n,l}^g, \preceq)$ of the Laplaciancoefficients, where $\mathcal{U}_{n,l}^g$ denote the set of $n$-vertexunicyclic graphs with the number of leaves $l$ and girth $g$. These results areused to disprove their conjecture. Moreover, the graphs with minimumLaplacian-like energy in $\mathcal{U}_{n,l}^g$ are also studied.

Laplacian coefficients of unicyclic graphs with the number of leaves and girth