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

Abstract

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 its Laplacian matrix. Motivated by Ilić and Ilić's conjecture [A. Ilić, M. Ilić, 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 ${\mathcal{U}}_{n,l}$ of all $n$-vertex unicyclic graphs with the number of leaves $l$, we investigate properties of the minimal elements in the partial set $({\mathcal{U}}_{n,l}^g,⪯)$ 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 Laplacian-like energy in ${\mathcal{U}}_{n,l}^g$ are also studied.

arXiv comment19 page, 4figures

References in corpus (1)

• On signless Laplacian coefficients of unicyclic graphs with given matching number