Regular and Strongly Regular Planar Graphs

We give a constructive proof that a planar graph on \( n \) vertices with degree of regularity \( k \) exists for all pairs \( (n,k) \) except for two pairs \( (7,4) \) and \( (14,5) \). We continue this theme by classifying all strongly regular planar graphs, and then consider a new class of graphs called \( 2 \)-\emph{strongly regular}. We conclude with a conjectural classification of all planar \( 2 \)-strongly regular graphs.