A note on the third invariant factor of the Laplacian matrix of a graph

Abstract

Let $G$ be a simple connected graph with $n\ge 5$ vertices. In this note, we will prove that $s_3(G)\le n$, and characterize the graphs which satisfy that $s_3(G)=n,n-1,n-2,$ or $n-3$, where $s_3(G)$ is the third invariant factor of the Laplacian matrix of $G$.

8 pages