Asymptotically approaching the Moore bound for diameter three by Cayley graphs

Abstract

The largest order n(d, k) of a graph of maximum degree d and diameter k cannot exceed the Moore bound, which has the form M (d, k) = d k − O(d k−1 ) for d → ∞ and any fixed k. Known results in finite geometries on generalised (k + 1)-gons imply, for k = 2, 3, 5, the existence of an infinite sequence of values of d such that n(d, k) = d k − o(d k ). This shows that for k = 2, 3, 5 the Moore bound can be asymptotically approached in the sense that n(d, k)/M (d, k) → 1 as d → ∞; moreover, no such result is known for any other value of k ≥ 2. The corresponding graphs are, however, far from vertex-transitive, and there appears to be no obvious way to extend them to vertex-transitive graphs giving the same type of asymptotic result.The second and the third author (2012) proved by a direct construction that the Moore bound for diameter k = 2 can be asymptotically approached by Cayley graphs. Subsequently, the first and the third author (2015) showed that the same construction can be derived from generalised triangles with polarity.By a detailed analysis of regular orbits of suitable groups of automorphisms of graphs arising from polarity quotients of incidence graphs of generalised quadrangles with polarity, 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 Moore bound for diameter 3 can thus as well be asymptotically approached by Cayley graphs. We also show that this method does not extend to constructing Cayley graphs of diameter 5 from generalised hexagons with polarity.

On the nonexistence of lattice tilings of Zn by Lee spheres

“…, where exp(G) denotes the exponent of G and ζ exp(G) is a primitive exp(G)-th root of unity. At first glance, we do not have any obvious relation between χ(T ) and χ(T (2) ). However, by applying (·) (2) on T 2 = 2G − T (2) + 2n recursively, we get…”

“…This upper bound was also pointed by Dougherty and Faber in [8], in which they investigated the upper and lower bounds of AC(2d, k) by considering the associated lattice tilings of Z n by Lee spheres. Beside giving a better upper bound on AC(∆, k), it is also a challenging task to find better/exact lower bounds on AC(∆, k) by constructing special Cayley graphs; for recent progress on this topic as well as the same problem for nonabelian Cayley graphs, we refer to [1], [2], [20], [21], [25] and [33].…”

“…First, let us look at the existence of T in G/H which is isomorphic to the cyclic group C 5 of order 5. Changing T to T (2) , note that T (4) = T , we get that (T (2) ) 2 ≡ −T (4) + 2n = −T + 2n (mod G/H).…”

“…and T (3) . By using MAGMA [5], we calculate the resultant f 12 of f 1 and f 2 as well as the resultant f 13 of f 1 and f 3 with respect to T (2) . Then we compute the resultant g of f 12 and f 13 with respect to T (2 2 ) .…”

“…By the symmetric property of T , T (2) and T (3) , it also holds if we replace T in it by T (2) or T (3) .…”