  The modular graph $C_{a,b,c,d}$ on the torus is a three loop planar graph in
which two of the vertices have coordination number four, while the others have
coordination number two. We obtain an eigenvalue equation satisfied by
$C_{a,b,c,d}$ for generic values of $a,b,c$ and $d$, where the source terms
involve various modular graphs. This is obtained by varying the graph with
respect to the Beltrami differential on the toroidal worldsheet. Use of several
auxiliary graphs at various intermediate stages of the analysis is crucial in
obtaining the equation. In fact, the eigenfunction is not simply $C_{a,b,c,d}$
but involves subtracting from it specific sums of squares of non--holomorphic
Eisenstein series characterized by $a,b,c$ and $d$.
