  We study the chromatic symmetric functions of graph classes related to
spiders, namely generalized spider graphs (line graphs of spiders), and what we
call horseshoe crab graphs. We show that no two generalized spiders have the
same chromatic symmetric function, thereby extending the work of Martin, Morin
and Wagner. Additionally, we establish that a subclass of generalized spiders,
which we call generalized nets, has no e-positive members, providing a more
general counterexample to the necessity of the claw-free condition. We use yet
another class of generalized spiders to construct a counterexample to a problem
involving the $e$-positivity of claw-free, P4-sparse graphs, showing that
Tsujie's result on the e-positivity of claw-free, P4-free graphs cannot be
extended to graphs in this set. Finally, we investigate the e-positivity of
another type of graphs, the horseshoe crab graphs (a class of unit interval
graphs), and prove the positivity of all but one of the coefficients. This has
close connections to the work of Gebhard and Sagan and Cho and Huh.
