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I have no experience in graph theory and am a little confused with how Hugh Osborn represents a perturbation expansion with diagrams on page 15 of these notes.

We have a perturbation expansion

enter image description here

My questions are: 1. In the diagrammatic interpretation part, 2 points are joined to create one line. So shouldn't a vertex with $k$ indices be at the intersection of $k/2$ lines? 2. If $A_{ij}^{-1}$ is a straight line from $i$ to $j$, how do we have a vertex with only one line? 3. Where do the points $i,j,k,l$, etc. go on the loops?

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  • $\begingroup$ Related post by OP: physics.stackexchange.com/q/492098/2451 $\endgroup$ – Qmechanic Jul 19 at 10:33
  • $\begingroup$ @Qmechanic This one is from the same set of notes. There, I'm asking about how to evaluate a series algebraically by differentiating. Here, I'm asking about how the resulting expansion corresponds to the diagrams shown. $\endgroup$ – saad Jul 19 at 10:38
  • $\begingroup$ I find the questions hard to understand. Have you try to draw the diagrams from 1.133 by yourself? For example, the first term is a vertex with two legs, that you can connect with a line, so you get the first bubble. $\endgroup$ – Adam Jul 19 at 11:20
  • $\begingroup$ $A_{ij}^{-1}$ corresponds to points $i$ and $j$ connected by a line. How do I create a vertex V_{ij}out of this straight line? $\endgroup$ – saad Jul 19 at 11:31
  • $\begingroup$ It works the other way around. Starting from $V_{ij}A^{-1}_{ij}$, you can draw a picture by drawing $V_{ij}$ as a dot with two legs (one with label $i$, the other with label $j$) connected by a line representing $A^{-1}_{ij}$. $\endgroup$ – Adam Jul 19 at 12:02
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Note that the reason why you have bubbles is that you have many repeated (i.e. dummy) indices. In the first diagram for example, i and j label the two ($k=2$) lines joined by $-V_{ij}$, but they also label the points joined by $A^{-1}_{ij}$, so that is why the two lines close to form a bubble.

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  • $\begingroup$ I did not know that indices corresponded to lines and not points. This clears up a lot. Thank You! $\endgroup$ – saad Jul 19 at 15:41

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