### 4.3 Connectivity

MCT models characterize a field’s connectivity by quantifying the amount of flux interconnecting each
pair of photospheric sources. This is not possible in either pointwise mapping models or submerged poles
models. The domain graph of a field with domains and sources has edges connecting
vertices (Longcope, 2001). This defines an incidence matrix which is 1 when domain connects to
source , and 0 otherwise. The domain matrix relates domain fluxes to source fluxes ,
These relations
leave only domain fluxes undetermined (Longcope, 2001). Sweet’s configration, with
and , thus has only one domain flux not fixed by the sources. This one degree of freedom is set by
the flux passing through its one separator, which can be varied through reconnection without changing any
source fluxes. It can be shown in general that relation (18) augmented with the fluxes through
all separators uniquely determines all domain fluxes (Longcope and Klapper, 2002). A direct
consequence of this is that domain fluxes can only change from changing source fluxes, due to
emergence or submergence, or from transferring field lines across separators, through separator
reconnection.
There is not yet a method for enumerating the domains in a general MCT field, however, Longcope and
Klapper (2002) present a method for enumeration in potential fields and those topologically equivalent to
them, and Beveridge and Longcope (2005) generalize it. A field with separators, nulls, with
unbroken fans, and sources has domains, including those in the mirror
corona.
Assuming there are no upright nulls (which introduce such complications as fans and separators in the
photosphere) we set the number of coronal separators to and subtract off half of the domains
not present in the footprint (the number of footprints is given by Equation [17]) to find the total number of
domains not counting those in the mirror corona. Designating by the number of coronal nulls
gives (Beveridge and Longcope, 2005)

once has been eliminated using Equation (16). Furthermore, the number of purely coronal domains,
those without footprints, is , equal to the number of independent circuits formed
by coronal separators (Longcope and Klapper, 2002). This is a consequence of the fact that each purely
coronal domain must be engirdled by a unique circuit of separators.