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    $\begingroup$ When you describe a problem abstractly, using only mathematical symbolism, it's essential that you be perfectly correct about it. You lost me at the point you redefined "$X_{ij}$" because evidently it's a pair of subscripts (that what an "arg min" gives) but you don't seem to be employing it that way. I also found the sequence "$j\ne k\ne i$" confusing because it's ambiguous. Consider, please, explaining your question more clearly by describing what you're doing in terms of the graph (or its adjacency matrix) and/or including a small illustrated example. $\endgroup$ Commented Jan 16 at 15:47
  • $\begingroup$ @whuber Added more explanation and an illustrated example $\endgroup$ Commented Jan 16 at 16:06
  • $\begingroup$ What is $|E|$? A determinant? Is $E$ a matrix? Only half of it is defined. $\endgroup$ Commented Jan 16 at 16:53
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    $\begingroup$ @AdamO $|E|$ is the cardinal or size of set $E$. $\endgroup$ Commented Jan 16 at 17:46
  • $\begingroup$ With such a question my first step would be to simulate a few small cases. Then try to construct some cases where it might most strongly seem to risk failure (you will need a better understanding of the problem than I do now to figure out a good way to do that). If it's still looking like it holds for all those you can think of, then start looking for a theorem / asking around about existing results. Usually, however, such an exploration, carried out with a little care, is sufficient to disprove most such guesses. It can save a lot of time spent trying to prove false theorems. $\endgroup$ Commented Jan 17 at 0:39