pentagon and one square

As we have discussed, a star graph corresponds to $(n-2)!$ multi-peripheral Feynman diagrams, each characterized by a permutation of $n-2$ particles. For star graph, the vertices of the corresponding polytope are $(n-2)!$ multi-peripheral Feynman diagrams, each characterized by a permutation of $n-2$ particles. Two vertices are connected by an edge iff their permutations differ by a relabeling of two adjacent particles. It is known that a polytope with such vertices is the so-called [*permutohedron*](https://en.wikipedia.org/wiki/Permutohedron) [@permutohedron] living in $n-3$ dimensions, which we denoted as ${\mathcal{P}}_{n-2}$ (in usual literature, it is called $P_{n-1}$). Therefore. we have mapped a star graph to a star graph to a permutohedron, and let’s give some explicit examples.

For $n=4$, ${\rm PT}(1,2,3,4)=C^S_4(2)$, so the associahedron is also a permutohedron, but in a different view, ${\mathcal{P}}_1$ , corresponding to the permutation of $1,3$.

For $C^S_5(1)$, see , it’s mapped to ${\mathcal{P}}_2$ ,see figure \[per of S5\] . The leg 1 and 5 in each Feynman diagram are particular and all other legs take part in the permutations. Any two adjacent vertices, which differs by a relabeling of two adjacent particles, ,share a common pole represented by an edge.

at (5,0) [ ]{}; (-1,-1.25)–(1,0)–(3,-1.25)–(3,-3.3)–(1,-4.55)–(-1,-3.3)–(-1,-1.25); at 