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We have introduced the signature of Feynman Graph as: $$ \omega_D(\Gamma)=D|\Gamma|-2E_\Gamma-V_\Gamma $$ in which $E_\Gamma$ is the number of the edges, $V_\Gamma$ is the number of vertexes and $D$ is the number of dimensions of the field theory corresponding to the graph under consideration.

Is this signature conserved for graphs with the same characteristics?

We have been given as an exercise to prove that indeed it is in the special case of $D=4$, with the values:

  • +0 to a four-valent vertex
  • -1 to a three-valent vertex
  • +2 to an internal edge

and, under those hypothesis, $\omega_4$ should be zero if the graph has 4 external edges, +1 if has 3 external edges and 0 if it has 2 external edges.

My idea was to consider a graph without external edges and try to calculate its signature, deriving then the signature for inserting 3- and 4-valent vertexes to create external edges. However, the first part has proven though than I thought. Has anyone any idea on how to mage the problem?

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  • $\begingroup$ of course, my apology $\endgroup$
    – Drebin J.
    Commented Jan 29, 2018 at 16:26
  • $\begingroup$ What do you exactly mean by "Is this signature conserved for graphs with the same characteristics?"? In what sense "conserved"? And what does "same characteristics" mean? That the graphs are isomorphic? Or something weaker, such as they having the same order/size/etc.? $\endgroup$ Commented Jan 29, 2018 at 16:33
  • $\begingroup$ What is a graph "with the same characteristics"? $\endgroup$
    – knzhou
    Commented Jan 29, 2018 at 16:39
  • $\begingroup$ I mean: is the signature the same for graphs which have, for example, the same number of external edges, built with the same n-valent vertexes and so on? $\endgroup$
    – Drebin J.
    Commented Jan 29, 2018 at 16:58
  • $\begingroup$ @DrebinJ. well your definition only depends on the valency of the vertex, and the number of edges of each type in the graph. So I would say it is by definition the same, unless I misunderstood something in your question. $\endgroup$ Commented Jan 29, 2018 at 21:36

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