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arXiv · 1311.1987

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

Jie Zhang·Xiao-Dong Zhang
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Abstract

Let $G$ be a graph of order $n$ and let $\mathcal{L}(G,λ)=\sum_{k=0}^n (-1)^{k}c_{k}(G)λ^{n-k}$ 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 $(\mathcal{U}_{n,l}^g, \preceq)$ 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.

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Jie Zhang, Xiao-Dong Zhang. 2013-11-08. Laplacian coefficients of unicyclic graphs with the number of leaves and girth. https://arxiv.org/abs/1311.1987

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