The bounds of the energy and Laplacian energy of chain graphs

Abstract

Let $G$ be a simple connected graph of order $n$ with $m$ edges. The energy $\varepsilon(G)$ of $G$ is the sum of the absolute values of all eigenvalues of the adjacency matrix $A$. The Laplacian energy is defined as $LE(G)=\sum_{i=1}^{n}|\mu_{i}-\frac{2m}{n}|$, where $\mu_{1},\mu_{2},\dots,\mu_{n}$ are the Laplacian eigenvalues of a graph $G$. In this article, we obtain some upper and lower bounds on the energy and Laplacian energy of chain graph. Finally, we attain the maximal Laplacian energy among all connected bicyclic chain graphs by comparing algebraic connectivity.

Keywords

• chain graph
• energy
• laplacian energy
• vertex cover number
• algebraic connectivity