$2,3,4$.

A more non-trivial example for $n=6$. According to = + + + , we have the 18 Feynman diagrams \[example6pt\] T() &=& { , , , , } && { , } && { , , , , , } . Here ${\color{blue}\big(}{\color{blue}\text{{ 1}}}\leftrightarrow {\color{blue}\text{{ 3}}}{\color{blue}\big)}$ means five more Feynman diagrams owing to the symmetry of $1$ and $3$ in the labelled tree.

### Hamiltonian graph , Star graph and beyond {#hamiltonian-graph-star-graph-and-beyond .unnumbered}

Above we have seen examples , for Hamiltonian graph, which is the so-called Parke-Taylor graph. All connected line segment except the labelled tree itself correspond to a pole and all compatible $n-3$ connected line segment from these correspond to a pole Feynman diagrams. The exact pole sets and Feynman diagram sets are given by \[p of H\] P([PT]{}(1,2,...,n))&=&{s\_[i,i+1,...,j]{}| 1i&lt;j n-1,  (i,j)=(1,n-1)}\
T([PT]{}(1,2,...,n))&=&{}. The number of poles are $\frac{n(n-3)}{2}$. The number of Feynman diagrams ${\rm Cat}_{n-2}$ can be seen from the recursion which gives the recursion of Catalan numbers directly.

The ${\rm PT}(1,2,3,4)$ in can also be seen as $C^S_4(2)$, \[example4pt22\] T( ) = { , }, which are two multi-peripheral Feynman diagrams with the permutations of $1,3$. Another example for star graph is given in . 