Let G be a graph of order n and let L(G,Ξ»)=βk=0nβ(β1)kckβ(G)Ξ»nβk be the characteristic polynomial of its
Laplacian 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 or
vertices of degree two, Linear Algebra and its Applications
431(2009)2195-2202.] on all extremal graphs which minimize all the Laplacian
coefficients in the set Un,lβ of all n-vertex unicyclic graphs
with the number of leaves l, we investigate properties of the minimal
elements in the partial set (Un,lgβ,βͺ―) of the Laplacian
coefficients, where Un,lgβ denote the set of n-vertex
unicyclic graphs with the number of leaves l and girth g. These results are
used to disprove their conjecture. Moreover, the graphs with minimum
Laplacian-like energy in Un,lgβ are also studied.Comment: 19 page, 4figure