ZERO-SYMMETRIC GRAPHS OF GIRTH 3

This chapter discusses 0-symmetric graphs of girth 3. The chapter presents proof for a theorem that states that a graph of type 3 T or 3 Z cannot have girth 3. The only 0-symmetric graphs of girth 3 are of type 1 Z. They can be obtained by a procedure that might be described as “blowing up the vertices of a 1-regular graph to triangles.” In any trivalent graph G with 2n vertices, one can replace the vertices by triangles, thus obtaining a trivalent graph tG with 6n vertices. If the original graph G belongs to class S, the derived graph tG is still vertex-transitive but either of class T or of class Z. An inspection of any s-arc of the symmetrical graph G and its augmented arc in tG shows that if G is s-regular, the derived graph tG is t-symmetric with t = s - 1, and this holds still in the simplest case s = 1, thus giving a 0-symmetric graph tG of girth 3 when blowing up a 1-regular graph G.