LAPLACIAN COEFFICIENTS OF UNICYCLIC GRAPHS WITH THE NUMBER OF LEAVES AND GIRTH

Motivated by Ilic and Ilic's conjecture [A. ILIC, M. ILIC: Laplacian coefficients of trees with given number of leaves or vertices of degree two. Linear Algebra Appl., 431 (2009), 2195-2202.], we investigate properties of the minimal elements in the partial set (U-n,l(g),<=) of the Laplacian coefficients, where U-n,l(g) 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 U-n,l(g) are also studied.