Using the replica method, we develop an analytical approach to compute the characteristic function for the probability P_N(K,λ) that a large N × N adjacency matrix of sparse random graphs has K eigenvalues below a threshold λ. The method allows to determine, in principle, all moments of P_N(K,λ), from which the typical sample to sample fluctuations can be fully characterized. For random graph models with localized eigenvectors, we show that the index variance scales linearly with N ≫ 1 for |λ|

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... 0, with a model-dependent prefactor that can be exactly calculated. Explicit results are discussed for Erdös-Rényi and regular random graphs, both exhibiting a prefactor with a non-monotonic behavior as a function of λ. These results contrast with rotationally invariant random matrices, where the index variance scales only as N, with an universal prefactor that is independent of λ. Numerical diagonalization results confirm the exactness of our approach and, in addition, strongly support the Gaussian nature of the index fluctuations.