Given a Coxeter group
W
W
, a
W
W
-graph
Γ
\Gamma
encodes a module
M
Γ
M_{\Gamma }
for the associated Iwahori-Hecke algebra
H
\mathcal {H}
. The strongly connected components of
Γ
\Gamma
, known as cells, are also
W
W
-graphs, and their modules occur as subquotients in a filtration of
M
Γ
M_{\Gamma }
. Of special interest are the
W
W
-graphs and cells arising from the Kazhdan-Lusztig basis for the regular representation of
H
\mathcal {H}
. We define a
W
W
-graph to be admissible if, like the Kazhdan-Lusztig
W
W
-graphs, it is edge-symmetric, bipartite, and has nonnegative integer edge weights. Empirical evidence suggests that for finite
W
W
, there are only finitely many admissible
W
W
-cells. We provide a combinatorial characterization of admissible
W
W
-graphs, and use it to classify the admissible
W
W
-cells for various finite
W
W
of low rank. In the rank two case, the nontrivial admissible cells turn out to be
A
A
-
D
D
-
E
E
Dynkin diagrams.