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We introduce the weighted graph Laplacian and the notion of Schr\"odingeroperator on a locally finite weighted graph. Concerning essentialself-adjointness, we extend Wojciechowski's and Dodziuk's results for graphswith vertex constant weight. The main result in this work states that on anymetrically complete weighted graph with bounded degree, the weighted graphLaplacian is essentially self-adjoint and the same holds for the Schr\"odingeroperator provided the associated quadratic form is bounded from below. Weconstruct for the proof a strictly positive and harmonic function which allowsus to write any Schr\"odinger operator as a weighted graph Laplacian modulo aunitary transform.

Essential self-adjointness for combinatorial Schr\"odinger operators I- Metrically complete graphs