  The vertex model is a popular framework for modelling tightly packed
biological cells, such as confluent epithelia. Cells are described by convex
polygons tiling the plane and their equilibrium is found by minimizing a global
mechanical energy, with vertex locations treated as degrees of freedom. Drawing
on analogies with granular materials, we describe the force network for a
localized monolayer and derive the corresponding discrete Airy stress function,
expressed for each $N$-sided cell as $N$ scalars defined over kites covering
the cell. We show how a torque balance (commonly overlooked in implementations
of the vertex model) requires each internal vertex to lie at the orthocentre of
the triangle formed by neighbouring edge centroids. Torque balance also places
a geometric constraint on the stress in the neighbourhood of cellular
trijunctions, and requires cell edges to be orthogonal to the links of a dual
network that connect neighbouring cell centres and thereby triangulate the
monolayer. We show how the Airy stress function depends on cell shape when a
standard energy functional is adopted, and discuss implications for
computational implementations of the model.
