The first appearance of graphs

In the last section we calculated some 2 and 4-point functions:

$$\< v^1,v^2\> = A^{-1}_{1,2}$$

$$\< v^1,v^1\> = A^{-1}_{1,1}$$

$$\< v^1,v^2,v^3,v^4\> = A^{-1}_{2,3}A^{-1}_{1,4} + A^{-1}_{2,4}A^{-1}_{1,3} + A^{-1}_{3,4}A^{-1}_{1,2}$$

$$\< v^1,v^1,v^3,v^4\> = 2A^{-1}_{1,4}A^{-1}_{1,3} + A^{-1}_{3,4}A^{-1}_{1,1}$$

$$\< v^1,v^1,v^1,v^4\> = 3A^{-1}_{1,4}A^{-1}_{1,1}$$

$$\< v^1,v^1,v^4,v^4\> = 2A^{-1}_{1,4}A^{-1}_{1,4} + A^{-1}_{4,4}A^{-1}_{1,1}$$

$$\< v^1,v^1,v^1,v^1\> = 3A^{-1}_{1,1}A^{-1}_{1,1}~.$$

It is convenient to represent each of products appearing on the right as a  graph, where the vertices represent the indices of the coordinates $v^i$ appearing in the $m$-point function, and each $A^{-1}_{1,j}$ becomes an edge from vertex $i$ to vertex $j$.