A Note on a Geometric Construction of Large Cayley Graphs of given Degree and Diameter

An infinite series and some sporadic examples of large Cayley graphs with given degree and diameter are constructed. The graphs arise from arcs, caps and other objects of finite projective spaces. A simple finite graph Γ is a (∆, D)-graph if it has maximum degree ∆, and diameter at most D. The (∆, D)-problem (or degree/diameter problem) is to determine the largest possible number of vertices that Γ can have. Denoted this number by n(∆, D), the well-known Moore bound states that n(∆, D) ≤ ∆(∆−1) −2 ∆−2 . This is known to be attained only if either D = 1 and the graph is K∆+1, or D = 2 and ∆ = 1, 2, 3, 7 and perhaps 57. If in addition Γ is required to be vertex-transitive, then the only known general lower bound is given as n(∆, 2) ≥ ⌊∆+ 2 2 ⌋ · ⌈∆+ 2 2 ⌉ . (1) This is obtained by choosing Γ to be the Cayley graph Cay(Za × Zb, S), where a = b 2 c, b = d ∆+2 2 e, and S = { (x, 0), (0, y) | x ∈ Za\{0}, y ∈ Zb\{0} }. If ∆ = kD+m, where k,m are integers and 0 ≤ m < D, then a straightforward generalization of this construction results in a Cayley (∆, D)-graph of order ⌊∆+D D ⌋D−m · ⌈∆+D D ⌉m . (2) Received by the editors: 05.11.2008. 2000 Mathematics Subject Classification. 05C62, 51E21.