Polarity graphs revisited

Abstract

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 byŠiagiová and the second author [J.

Asymptotically approaching the Moore bound for diameter three by Cayley graphs

“…For further analysis we will need a description of the neighbourhood N(u) of a few vertices u of A(q), that is, the set of all v ∈ V adjacent to u in A(q). By the adjacency rule in A(q), a vertex u = [x] is adjacent to precisely the vertices v = [y] = [x] that are points on the line π[x] spanned by the vectors in (1). Informally, a projective vector [x] is adjacent in A(q) precisely to the projective vectors in π[x] distinct from x.…”

“…Unfortunately, by the classification of maximal subgroups of the Ree groups [13,12], the group Re(q) = 2 G 2 (q), a simple group of order q 3 (q 3 + 1)(q − 1), does not contain a subgroup of order O(q 5 ) for q → ∞. The approach that works for construction of large Cayley graphs of diameter 2 from generalised triangles in [1], and of diameter 3 from generalised quadrangles in this article, thus does not carry over to an analogous construction for diameter 5 from generalised hexagons.…”

“…This result was obtained in [22] by a direct construction of certain Cayley graphs of one-dimensional affine groups over finite fields of characteristic 2. Later in [1] it was shown that the construction of [22] is equivalent to extending a regular orbit of a polarity quotient of the incidence graph of a generalised triangle under the action of a suitable group.…”

“…In fact, we prove that for an infinite set of values of d there exist Cayley graphs of degree d, diameter 3, and order d 3 − O(d 2.5 ). The method is a variant of the one used in [1], namely, extension of a regular orbit of a suitable subgroup of the automorphism group of a polarity quotient of the incidence graph of a generalised quadrangle. Details, however, are much more subtle and complex in comparison with those of [1].…”

“…The method is a variant of the one used in [1], namely, extension of a regular orbit of a suitable subgroup of the automorphism group of a polarity quotient of the incidence graph of a generalised quadrangle. Details, however, are much more subtle and complex in comparison with those of [1]. We also show that an extension of this method is not feasible for proving an analogous result for diameter 5.…”