Small World Graphs

Node relationship graphs are depicted by matrices with $m$ edges and $n$ nodes. Containing only -1 and 1 in every row they are incidence matrices.

$$A=\left[\begin{array}{ccc}-1& 1& \\ & 1& -1\\ 1& & -1\end{array}\right]$$

$A$ depicts a graph of three nodes (columns) being connected by three edges (rows). $r=2$

Properties of Loops in the Graph

Geometrically loops form whenever a group of edges connecting nodes is dependent.

Euler’s Relation

$$\text{no. nodes}-\text{no. edges}+\text{no. loops}=1$$