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In this work all the dessins d'enfant with no more than 4 edges are listedand their Belyi pairs are computed. In order to enumerate all dessins thetechnique of matrix model computations was used. The total number of dessins is134; among them 77 are spherical, 53 of genus 1 and 4 of genus 2. The orders ofautomorphism groups of all the dessins are also found.  Dessins are listed by the number of edges. Dessins with the same number ofedges are ordered lexicographically by their lists of 0-valencies. Thecorresponding matrix model for any list of 0-valencies is given and computed.Complex matrix models for dessins with 1 -- 3 edges are used. For the dessinswith 4 edges we use Hermitian matrix model, correlators for which are computedin [1].

Catalog of dessins d'enfants with \le 4 edges