Abstract
The modular graph Ca,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 Ca,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 Ca,b,c,d but involves subtracting from it specific sums of squares of non-holomorphic Eisenstein series characterized by a, b, c and d.
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