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Using periodic-orbit theory beyond the diagonal approximation we investigatethe form factor, $K(\tau)$, of a generic quantum graph with mixing classicaldynamics and time-reversal symmetry. We calculate the contribution from pairsof self-intersecting orbits that differ from each other only in the orientationof a single loop. In the limit of large graphs, these pairs produce acontribution $-2\tau^2$ to the form factor which agrees with random-matrixtheory.Comment: 5 pages (2 figs). Version to appear in PRL [RevTeX4] - with a bonus  appendix for nlin.CD user

Leading Off-Diagonal Correction to the Form Factor of Large Graphs