Lemma 55.5.2. Classification of proper subgraphs of the form

$\xymatrix{ \bullet \ar@{-}[r] & \bullet \ar@{-}[r] & \bullet }$

If $n > 3$, then given a triple $i, j, k$ of $(-2)$-indices with at least two $a_{ij}, a_{ik}, a_{jk}$ nonzero, then up to ordering we have the $m$'s, $a$'s, $w$'s

1. are given by

$\left( \begin{matrix} m_1 \\ m_2 \\ m_3 \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & 0 \\ w & -2w & w \\ 0 & w & -2w \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ w \end{matrix} \right)$

with $2m_1 \geq m_2$, $2m_2 \geq m_1 + m_3$, $2m_3 \geq m_2$, or

2. are given by

$\left( \begin{matrix} m_1 \\ m_2 \\ m_3 \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & 0 \\ w & -2w & 2w \\ 0 & 2w & -4w \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ 2w \end{matrix} \right)$

with $2m_1 \geq m_2$, $2m_2 \geq m_1 + 2m_3$, $2m_3 \geq m_2$, or

3. are given by

$\left( \begin{matrix} m_1 \\ m_2 \\ m_3 \end{matrix} \right), \quad \left( \begin{matrix} -4w & 2w & 0 \\ 2w & -4w & 2w \\ 0 & 2w & -2w \end{matrix} \right), \quad \left( \begin{matrix} 2w \\ 2w \\ w \end{matrix} \right)$

with $2m_1 \geq m_2$, $2m_2 \geq m_1 + m_3$, $m_3 \geq m_2$.

Proof. See discussion above. $\square$

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