Vertex weighted Laplacian Energy of union of graphs

Abstract

The vertex weighted Laplacian energy with respect to the vertex weight $w$ of a graph $G$ with $n$ vertices is defined as ~$LE_w(G)=\sum\limits_{i=1}^n|\mu_i-\bar{w}|$, where ${{\mu }_{1}},{{\mu }_{2}},...,{{\mu }_{n}}$ are the Laplacian eigenvalues of $G$ and $\bar{w}$ is the average value of the weight $w$. In this paper, we derive upper and lower bounds of weighted Laplacian energy of union of $k$-number of connected disjoint graphs $G_1$, $G_2$,...,$G_k$ and hence consider some particular cases.

Keywords

• Eigenvalue
• Energy (of graph)
• Laplacian energy