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

TL;DR

This paper investigates the Laplacian coefficients of unicyclic graphs with fixed leaves and girth, disproving a previous conjecture and analyzing graphs with minimal Laplacian-like energy.

Contribution

It provides new insights into the extremal properties of Laplacian coefficients in unicyclic graphs, challenging existing conjectures.

Findings

Disproved Ilić and Ilić's conjecture on Laplacian coefficients.

Characterized minimal Laplacian coefficient graphs within specified classes.

Analyzed graphs with minimal Laplacian-like energy.

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 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 $\mathcal{U}_{n,l}$ 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 Laplacian-like energy in $\mathcal{U}_{n,l}^g$ are also studied.