* Trees
- a connected, undirected network with no self-avoiding loops
- if all components of a network are trees, it is called a forest
- trees are drawn in a rooted manner, vertices at the bottom are leaves
- any vertex could be used as a root, including a leaf
- river network, actual tree
- important in the theoretical study of networks, also used as data
structures
- there is exactly one path between any given pair
- tree on n vertices has exactly n-1 edges, and a network with n-1 edges
is necessarily a tree
* Directed acyclic graphs (DAGs)
- directed networks with no loops (no self-loops too)
- WWW is not acyclic, but a citation network is.
- a DAG can always be drawn such that all edges point-downwards!
1. find a vertex with no-outgoing edges
2. no such vertex => graph is cyclic. Else, remove it with all its
edges
3. if all vertices have been removed, the network is acyclic, else
go to 1
- adjacency matrix is strictly upper triangular => eigenvalues are zero
- if all eigenvalues are zero, the network is acyclic:
- use L_r = sum(lambda_i^r)
- such matrices are also called nilpotent matrices
* Planar networks
- a network that can be drawn on a plane without having any edges cross
- almost always planar networks can be drawn so that some edges cross.
Planarity only means that at least one arrangement exists in which
edges don't cross
- Most networks are not planar because there is no relevant 2d space in
which they are embeded
- but some are:
- trees, river networks
- road network (almost planar)
- shared border between the regions
- chromatic number and the four-color theorem