  Poly(indenoindenes) are {\pi}-conjugated ladder carbon polymers with
alternating hexagons and pentagons hosting one unpaired electron for each
five-membered ring in the open-shell limit. Here we study the main magnetic
interactions that are present in finite oligo(indenoindenes) (OInIn),
classifying the six possible isomers in two different classes of three isomers
each. One class can be rationalized by frustrated S = 1/2 Heisenberg chains,
with ferromagnetic interactions between neighbour sites and antiferromagnetic
interactions between the next neighbours. The other class is characterized by
the more trivial antiferromagnetic order. Employing several levels of theory we
further show that the ground state of one of the isomers is a valence-bond
solid (VBS) of ferromagnetic dimers (S = 1). This is topologically similar to
that of the Affleck-Kennedy-Lieb-Tasaki (AKLT) model, which is known to show
fractional S = 1/2 states at the edges.
