This paper presents a nearly-linear time algorithm for spectral sparsification of weighted graphs. Given a weighted graph G and a parameter ε > 0, the algorithm produces a weighted subgraph H with O(n log n / ε²) edges that approximates G in a spectral sense. Specifically, for all vectors x, the quadratic form of the Laplacian matrices of G and H are within a factor of (1 ± ε). This improves upon previous results, including those of Spielman and Teng, which had O(n log^c n) edges for some large constant c, and those of Benczúr and Karger, which only satisfied the condition for x in {0, 1}^n. A key component of the algorithm is a nearly-linear time subroutine that computes approximate effective resistances between any two vertices in a graph in O(log n) time. Effective resistance is defined as the potential difference between two vertices when a unit current is injected at one and extracted at the other. The algorithm samples edges from G with probabilities proportional to their effective resistances, and then constructs H by including each sampled edge with a weight adjusted to maintain the total weight of the original edges. The main result is that if the number of samples q is sufficiently large, then the quadratic forms of the Laplacian matrices of G and H are close. This is proven using a concentration inequality for symmetric rank 1 matrices. The algorithm also shows that using approximate effective resistances for sampling does not significantly affect the sparsifier's quality. The paper also demonstrates how to compute approximate effective resistances in nearly-linear time, which is essential for the algorithm. This is achieved using Spielman and Teng's nearly-linear time solver and the Johnson-Lindenstrauss Lemma. The algorithm can then compute the effective resistance between any two vertices in O(log n) time, which is used to construct the sparsifier. Finally, the paper shows that the sparsifier H has additional properties that make it useful for solving linear systems. Specifically, it ensures that the sum of the rescaled weights of edges incident to any vertex is bounded by twice the degree of that vertex. This property is important for the construction of ultrasparsifiers and for preconditioning in linear systems. The algorithm runs in nearly-linear time and produces a sparsifier with O(n log n / ε²) edges that satisfies the spectral approximation condition with high probability.This paper presents a nearly-linear time algorithm for spectral sparsification of weighted graphs. Given a weighted graph G and a parameter ε > 0, the algorithm produces a weighted subgraph H with O(n log n / ε²) edges that approximates G in a spectral sense. Specifically, for all vectors x, the quadratic form of the Laplacian matrices of G and H are within a factor of (1 ± ε). This improves upon previous results, including those of Spielman and Teng, which had O(n log^c n) edges for some large constant c, and those of Benczúr and Karger, which only satisfied the condition for x in {0, 1}^n. A key component of the algorithm is a nearly-linear time subroutine that computes approximate effective resistances between any two vertices in a graph in O(log n) time. Effective resistance is defined as the potential difference between two vertices when a unit current is injected at one and extracted at the other. The algorithm samples edges from G with probabilities proportional to their effective resistances, and then constructs H by including each sampled edge with a weight adjusted to maintain the total weight of the original edges. The main result is that if the number of samples q is sufficiently large, then the quadratic forms of the Laplacian matrices of G and H are close. This is proven using a concentration inequality for symmetric rank 1 matrices. The algorithm also shows that using approximate effective resistances for sampling does not significantly affect the sparsifier's quality. The paper also demonstrates how to compute approximate effective resistances in nearly-linear time, which is essential for the algorithm. This is achieved using Spielman and Teng's nearly-linear time solver and the Johnson-Lindenstrauss Lemma. The algorithm can then compute the effective resistance between any two vertices in O(log n) time, which is used to construct the sparsifier. Finally, the paper shows that the sparsifier H has additional properties that make it useful for solving linear systems. Specifically, it ensures that the sum of the rescaled weights of edges incident to any vertex is bounded by twice the degree of that vertex. This property is important for the construction of ultrasparsifiers and for preconditioning in linear systems. The algorithm runs in nearly-linear time and produces a sparsifier with O(n log n / ε²) edges that satisfies the spectral approximation condition with high probability.