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For the following question:

We call an undirected graph a basic graph if its adjacency matrix is such that all diagonal entries are zero and all off-diagonal entries are less than or equal to one. Let $G = (V,E)$ be a basic graph. We call any set of three distinct vertices $\{u,v,w\} \in V$ a trinity, if each pair of vertices in the set is connected by an edge.

I need to find an expression for the number of trinitys (made up term for this question) contained in a basic graph, in terms of the trace of an appropriate matrix and ended up obtaining the expression trace(A³)/6.

Would this be correct?

Misha Lavrov
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ANM
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    This is not "graph theory question 1". There have actually been over 20000 questions on graph theory on this website already. I have changed your title to be more descriptive, and replaced the image by actual text, something you should also have done yourself. – Misha Lavrov Mar 30 '21 at 00:40
  • You are definitely right, I wasn't sure what to add for a title and since "Graph Theory Question 1" wasn't taken, I went with that. Thank you though :) – ANM Mar 30 '21 at 01:22

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Yes, the $(i,j)$ entry of $A^3$ counts the number of walks of length $3$ from $i$ to $j$, so $\operatorname{trace}(A^3)$ counts the number of directed $3$-cycles, distinguished by the starting node and orientation. Each trinity (commonly known as a triangle or $3$-clique) is counted $3!=6$ times in this formula, so divide by $6$.

RobPratt
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