## Journal of Applied Mathematics

- J. Appl. Math.
- Volume 2012 (2012), Article ID 404067, 11 pages.

### On the Laplacian Coefficients and Laplacian-Like Energy of Unicyclic Graphs with $n$ Vertices and $m$ Pendent Vertices

#### Abstract

Let $\mathrm{\Phi}(G,\lambda )=\text{d}\text{e}\text{t}(\lambda {I}_{n}-L(G))={\sum}_{k=0}^{n}(-1{)}^{k}{c}_{k}(G){\lambda}^{n-k}$ be the characteristic polynomial of the Laplacian matrix of a graph $G$ of order $n$. In this paper, we give four transforms on graphs that decrease all Laplacian coefficients ${c}_{k}(G)$ and investigate a conjecture A. Ilić and M. Ilić (2009) about the Laplacian coefficients of unicyclic graphs with $n$ vertices and $m$ pendent vertices. Finally, we determine the graph with the smallest Laplacian-like energy among all the unicyclic graphs with $n$ vertices and $m$ pendent vertices.

#### Article information

**Source**

J. Appl. Math., Volume 2012 (2012), Article ID 404067, 11 pages.

**Dates**

First available in Project Euclid: 2 January 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.jam/1357153535

**Digital Object Identifier**

doi:10.1155/2012/404067

**Mathematical Reviews number (MathSciNet)**

MR2984200

**Zentralblatt MATH identifier**

1264.05082

#### Citation

Pai, Xinying; Liu, Sanyang. On the Laplacian Coefficients and Laplacian-Like Energy of Unicyclic Graphs with $n$ Vertices and $m$ Pendent Vertices. J. Appl. Math. 2012 (2012), Article ID 404067, 11 pages. doi:10.1155/2012/404067. https://projecteuclid.org/euclid.jam/1357153535