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The nodal domains of eigenvectors of the discrete Schrodinger operator onsimple, finite and connected graphs are considered. Courant's well known nodaldomain theorem applies in the present case, and sets an upper bound to thenumber of nodal domains of eigenvectors: Arranging the spectrum as a nondecreasing sequence, and denoting by $\nu_n$ the number of nodal domains of the$n$'th eigenvector, Courant's theorem guarantees that the nodal deficiency$n-\nu_n$ is non negative. (The above applies for generic eigenvectors. Specialcare should be exercised for eigenvectors with vanishing components.) The mainresult of the present work is that the nodal deficiency for genericeigenvectors equals to a Morse index of an energy functional whose value at itsrelevant critical points coincides with the eigenvalue. The association of thenodal deficiency to the stability of an energy functional at its criticalpoints was recently discussed in the context of quantum graphs[arXiv:1103.1423] and Dirichlet Laplacian in bounded domains in $R^d$[arXiv:1107.3489]. The present work adapts this result to the discrete case.The definition of the energy functional in the discrete case requires a specialsetting, substantially different from the one used in[arXiv:1103.1423,arXiv:1107.3489] and it is presented here in detail.

Stability of nodal structures in graph eigenfunctions and its relation to the nodal domain count