Minimal graphs of a torus, a projective plane and spheres and some properties of minimal graphs of homotopy classes

Contractible transformations of graphs consist of contractible gluing and deleting of vertices and edges of graphs. They partition all graphs into the family of homotopy classes. Contractible transformations do not change the Euler characteristic and the homology groups of graphs. In this paper we describe the minimal representatives of some homotopy classes and find the formula for computing the Euler characteristic of partite and some other graphs. We also describe the minimal graphs of a projective plane, a torus and a sphere.