Polarity graphs revisited

Polarity graphs, also known as Brown graphs, and their minor modifications are the largest currently known graphs of diameter 2 and a given maximum degree d such that d - 1 is a prime power larger than 5. In view of the recent interest in the degree-diameter problem restricted to vertex-transitive and Cayley graphs we investigate ways of turning the (non-regular) polarity graphs to large vertex-transitive graphs of diameter 2 and given degree. We review certain properties of polarity graphs, giving new and shorter proofs. Then we show that polarity graphs of maximum even degree d cannot be spanning subgraphs of vertex-transitive graphs of degree at most d + 2. If d - 1 is a power of 2, there are two large vertex-transitive induced subgraphs of the corresponding polarity graph, one of degree d - 1 and the other of degree d - 2. We show that the subgraphs of degree d - 1 cannot be extended to vertex-transitive graphs of diameter 2 by adding a relatively small non-edge orbital. On the positive side, we prove that the subgraphs of degree d - 2 can be extended to the largest currently known Cayley graphs of given degree and diameter 2 found by Siagiovia and the second author