
Mathematics 2013
Laplacian coefficients of unicyclic graphs with the number of leaves and girthAbstract: Let $G$ be a graph of order $n$ and let $\mathcal{L}(G,\lambda)=\sum_{k=0}^n (1)^{k}c_{k}(G)\lambda^{nk}$ 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)21952202.] on all extremal graphs which minimize all the Laplacian coefficients in the set $\mathcal{U}_{n,l}$ of all $n$vertex unicyclic graphs with the number of leaves $l$, we investigate properties of the minimal elements in the partial set $(\mathcal{U}_{n,l}^g, \preceq)$ of the Laplacian coefficients, where $\mathcal{U}_{n,l}^g$ 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 Laplacianlike energy in $\mathcal{U}_{n,l}^g$ are also studied.
