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I know that there exists a theorem that states the following. Graph $G$ contains a triangle if and only if there exist indices $i$ and $j$ so the both matrices $A_G$ and $A_G^2$ will have nonzero ($i,j$). However, I don't know the source of this theorem and I was unable to find it. Can you please give me a hint on where should I look for?

Rodrigo de Azevedo
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Keithx
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    Any graph theory text will have the well known theorem that the powers of the adjacency matrix counts the paths between vertices by their lengths. You may not need a formal reference for that theorem. – Ethan Bolker Dec 03 '23 at 17:00
  • The origin of looking at powers of adjacency matrices was addressed in this question, but this is interesting mainly to historians of math. When writing a proof, for example, you can either take this as a well-known fact, or cite almost any graph theory textbook, depending on the situation. – Misha Lavrov Dec 03 '23 at 18:24

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It's not really a "theorem": if $A_G$ (resp. $A^2_G$) have a non-zero values at $(i,j)$ it simply means there's a path of length 1 (resp. 2) between $i$ and $j$. Hence a triangle. The equivalence should write itself

Qise
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