Graphs are essential tools to illustrate relationships in given datasets visually. Therefore, generating graphs from another concept is very useful to understand it comprehensively. This paper will introduce a new yet simple method to obtain a graph from any finite affine plane. Some combinatorial properties of the graphs obtained from finite affine planes using this graph-generating algorithm will be examined. The relations between these combinatorial properties and the order of the affine plane will be investigated. Wiener and Zagreb indices, spectrums, and energies related to affine graphs are determined, and appropriate theorems will be given. Finally, a characterization theorem will be presented related to the degree sequences for the graphs obtained from affine planes.