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We consider a non self-adjoint Laplacian on a directed graph with non symmetric edge weights. We analyse spectral properties of this Laplacian under a Kirchhoff assumption. Moreover we establish isoperimetric inequalities in terms of the numerical range to show the absence of the essential spectrum of the Laplacian on heavy end directed graphs. 2010 Mathematics Subject Classification. 47A45, 47A12, 47A10, 47B25.doi:10.2298/fil1718671b fatcat:qybtkkl3yraflkmg4bmc4nrahu