Lemma 55.5.8. Classification of proper subgraphs of the form

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

Let $t > 5$ and $n > t$. Then given $t$ distinct $(-2)$-indices $i_1, \ldots , i_ t$ such that $a_{i_ ji_{j + 1}}$ is nonzero for $j = 1, \ldots , t - 1$, then up to reversing the order of these indices we have the $a$'s and $w$'s

1. are given by $w_{i_1} = w_{i_2} = \ldots = w_{i_ t} = w$, $a_{i_ ji_{j + 1}} = w$, and $a_{i_ ji_ k} = 0$ if $k > j + 1$, or

2. are given by $w_{i_1} = w_{i_2} = \ldots = w_{i_{t - 1}} = w$, $w_{j_ t} = 2w$, $a_{i_ ji_{j + 1}} = w$ for $j < t - 1$, $a_{i_{t - 1}i_ t} = 2w$, and $a_{i_ ji_ k} = 0$ if $k > j + 1$, or

3. are given by $w_{i_1} = w_{i_2} = \ldots = w_{i_{t - 1}} = 2w$, $w_{j_ t} = w$, $a_{i_ ji_{j + 1}} = 2w$, and $a_{i_{t - 1}i_ t} = 2w$, and $a_{i_ ji_ k} = 0$ if $k > j + 1$.

Proof. See discussion above. $\square$

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