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I am trying to apply the main theorem of this paper to a certain kind of graph and keep getting confused. The theorem uses $rank(Aut\Gamma)$ which is defined as "the number of $Aut \Gamma$ orbits on the set $V(\Gamma) \times V(\Gamma)$". ($\Gamma$ is a graph).

Now, my graph $\Gamma$ is built in this way: take a clique of $c$ vertices, labelled $\{1,2,\ldots,c\}$ and add $s=\binom{c-1}{2}$ additional vertices, each of which is connected to a different pair of two vertices from $\{2,\ldots,c\}$.

Question: What is $rank(Aut\Gamma)$?

My answer is 9 because the automorphism group is (apparently) $S_{c-1} \times S_{s}$ and there are 3 orbits for it. However, when I plug 9 into the theorem I get a contradiction with the rest of it (which involves objects I have a better grasp of so I am pretty sure I got the rest right).

Therefore, I suspect that my answer to the above question is wrong and I am in dire need of some enlightenment.

EDIT: Let's assume $c \geq 4$ to rule out sporadic cases.

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# Confused about orbits

I am trying to apply the main theorem of this paper to a certain kind of graph and keep getting confused. The theorem uses $rank(Aut\Gamma)$ which is defined as "the number of $Aut \Gamma$ orbits on the set $V(\Gamma) \times V(\Gamma)$". ($\Gamma$ is a graph).

Now, my graph $\Gamma$ is built in this way: take a clique of $c$ vertices, labelled $\{1,2,\ldots,c\}$ and add $s=\binom{c-1}{2}$ additional vertices, each of which is connected to a different pair of two vertices from $\{2,\ldots,c\}$.

Question: What is $rank(Aut\Gamma)$?

My answer is 9 because the automorphism group is (apparently) $S_{c-1} \times S_{s}$ and there are 3 orbits for it. However, when I plug 9 into the theorem I get a contradiction with the rest of it (which involves objects I have a better grasp of so I am pretty sure I got the rest right).

Therefore, I suspect that my answer to the above question is wrong and I am in dire need of some enlightenment.