2

I've been learning about the Laplacian of a graph, and all the cool things you can tell from it. For example, I know the smallest eigenvalue is always 0, and the mulitiplicity of that zero eigenvalue gives you the number of components in the graph. I noticed in some testing that the largest eigenvalue seems to be equal to the number of nodes in the largest component, but I can't find any articles/books that claim it is true or prove it so I think I just got a few luck cases. So my questions are:

1) Does the largest eigenvalue give the order of the largest connected component?

2) If not, can that number be found from the Laplacian?

Peter Mitrano
  • 123

1 Answers1

2

You mean $L:=D-A$ for undirected, unweighted simple graphs?

$1)$ No. For example, the greatest eigenvalue for the Laplacian of any $2n$-cycle is always $4$, and the greatest Laplacian eigenvalue of the path graph on $n$ vertices is strictly lesser than $4$ but tends to $4^-$ for huge $n$. The fact that $n$ figures as the greatest Laplacian eigenvalue of any complete or complete $r$-partite graphs (or any graph constructed by two graphs $G$ and $H$ by adding edges from all the nodes in $G$ to all the nodes in $H$) is no coincidence, though.

$2)$ There is an algorithm for identifying the connected components of any graph using its Laplacian matrix, so yes, albeit indirectly. Proceding as suggested here, one does find zero-one columns that encode which nodes of a graph belong to which connected components; adding the entries for each column gives the number of nodes in every component, then take the greatest.

Rócherz
  • 4,241