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There are a variety of notions of curvature that one could define on graphs. In this thesis, we introduce one of them which is called the ”Combinatorial Curvature”. This curvature is defined on graphs embedded on a surface, however as the name suggests,the combinatorial curvature of a vertex depends only on the combinatorial properties of the graph, and not on the geometric properties of the surface on which the graph is embedded. More precisely, the combinatorial curvature of a vertex is a function of the degree of the vertex and the faces which are incident to that vertex.In the main part of this thesis, we have tried to collect the most important results about the combinatorial curvature.... 

A moduli spase consists in classifying geometric objects up tp equivalence relations and its point are in 1-1 corrospondence with equivalence classes. One of the most important moduli spaces in algebraic geometry is «moduli space of stable n-pointed curves of genus zero» which has many applications in mathematics and theoretical physics.In [10] Knudsen shows that for every n 3 there is a smooth projective variety Xn that is a moduli space for stable n-pointed curves of genus zero.In this thesis we study moduli spaces and introduce Xn and its relation with cross ratios. This way we show Xn is smooth projective variety and a fine moduli space. [5] After that we study a paper by Sean Keel. [9]...