Algebraic connectivity of the second power of a graph

Abstract

Denote the Laplacian of a graph $G$ by $L(G)$ and its second smallest Laplacian eigenvalue by $λ_2(G)$. If $G$ is a graph on $n\ge 2$ vertices, then it is shown that the second smallest eigenvalue of $L(G) + \frac{1}{n} L(\overline{G^2})$ is at least 1, where $\overline{G^2}$ is the complement of the second power of $ G $. As a corollary of this result, it is shown that \begin{itemize} \item $ n \, λ_2(G) \ge λ_2(G^2), $ \item $ λ_2(G) \ge 1-\frac{|D_G|}{n}, $ \item $ λ_2(G) + λ_2(\Gb) \ge 1, $ \end{itemize} where $|D_G|$ is the number of vertices of eccentricity at least 3 in $G$.

Figures (1)

• Figure 1: The Graph $G_1$

Theorems & Definitions (2)

• Remark 2
• proof : Proof of Theorem \ref{['mainthm']}