Graph Theory and the Tree of Life

No one, to my knowledge, has ever applied the techniques of graph theory to examine the Tree of Life. Thus, being into graph theory as I am, I'm doing it.

A few definitions. A graph G is a collection of vertices (nodes) V(G), and a set of edges E(G). A node is generally represented by a dot or circle, and an edge by a line between two nodes. Thus the Tree is a graph, with each sephira as a node and each path as an edge.


Now, if we can find a set of nodes and edges between two nodes on a graph, then there is a path between them. If we can do this and then take a different set of edges and nodes to get back we have a cycle. A Hamiltonian Cycle is such a cycle that uses all node sin the graph only once (except the first/last node which is the same node).

The Tree of Life has a Hamiltonian cycle... which is also the Path of the Flaming Sword (among others) [note: lost my notes... again, not so sure on this]. Early inspections seem to indicate that all subgraphs (same Tree with removed sephira) have Hamiltonian cycles as well, which doesn't always happen.

there is a perfect matching on the Tree... ie, we can also pair off all the sephiroht without using an edge more than once. Obviously subgraphs don't have this property, as it requires an even number of nodes.

The Tree is 3/2-tough, which means that the best possible result we can get is by removing 3 nodes and yielding 2 disconnected components... remove Tiphareth and Netzach and Hod. Two parts with just Yesode and Malkuth and the rest in separate parts.

The Tree of Life is also 4-partite, meaning the nodes can be grouped into 4 parts and have no edges between any node and the other nodes in its part. Took me a minute to figure this one out, until i drew it in Word and started dragging around the nodes and the edges could move with me.

The girth of the graph (shortest cycle) is 3, as there is a triangle.

*sigh* I'll add more to this at another time, when I have more time for deeper study.