The Degree Diameter Problem for Cayley Graphs

Introduction
The degree/diameter problem for Cayley graphs can be stated as follows:

Given natural numbers d and k, find the largest possible number Nc(d,k) of vertices in a Cayley graph of maximum degree d and diameter k.

There are no better upper bounds for Nc(d,k) than the very general Moore bounds M(d,k)=(d(d-1)k-2)(d-2)-1. As an interesting fact, no Moore graph (graph whose order attains the Moore bound) is a Cayley graph. Indeed, neither the Petersen graph nor the Hoffman-Singleton graph are Cayley, and the possible Moore graph of degree 57 and diameter 2 is not even vertex-transitive.

Therefore, in attempting to settle the values of Nc(d,k), research activities in this problem follow the next two directions:


 * Increasing the lower bounds for Nc(d,k) by constructing ever larger graphs.


 * Lowering and/or setting upper bounds for Nc(d,k) by proving the non-existence of graphs whose order is close to the Moore bounds M(d,k).

Finding an upper bound for the general non-abelian case is still an open problem, while such an upper bound for the abelian case is already known.

Increasing the lower bounds for Nc(d,k)
The current largest lower bounds (of order close to d2/2) for Cayley graphs of diameter k=2 and an infinite set of values for the degree d>20, is given by Šiagiová and Širáň. Some lower bounds for trivalent graphs of diameter d>10 are given by Curtin.

Table of the orders of the largest known Cayley graphs for the undirected degree diameter problem
Below is the table of the largest known Cayley graphs (as of September 2009) in the undirected degree diameter problem for Cayley graphs of degree at most 3 ≤ d ≤ 20 and diameter 2 ≤ k ≤ 10. This table represents the best lower bounds known at present on the order of Cayley (d,k)-graphs. All optimal graphs are marked in bold. All Cayley graphs of order up to 33 are isomorphic to graphs in the lists available here, and some of the trivalent Cayley graphs of order up to 1000 are available here, but no information was given in these lists regarding diameter.

The following table is the key to the colors in the table presented above:

Lowering and/or setting upper bounds for Nc(d,k)
In attempting to set the values for Nc(d,k), most research efforts have been directed at the abelian case.

The abelian case

When the group considered is abelian, a Moore-like bound was given by Stanton and Cowan, and is asymptotically equivalent to (2k)d/2(d/2)!-1, where (d/2) is the number of generators from the group. Dougherty and Faber gave a list of optimal graphs for both the directed and undirected abelian Cayley case.

The non-abelian case

As said above, to set a Moore-like bound for the non-abelian case remains an open problem.

Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs
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 * $$d$$\$$k$$|| 2 || 3 || 4|| 5 ||  6 || 7 ||  8 ||  9 ||  10
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 * 5
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 * 7
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 * 8
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 * 9
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 * 10
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 * 10
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 * 11
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 * 12
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 * 13
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 * 14
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 * 15
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 * 16
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 * 17
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 * 18
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 * 19
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The following table is the key to the colors in the table presented above:

Cayley Graphs of Diameter Two
For the special case of diameter 2, Cayley graphs of degree up to 12 are known to be optimal. It is interesting to note that optimal Cayley graphs are smaller than the Moore bound (as shown in the table below).

Jana Šiagiová and Jozef Širáň have found a general construction for Cayley graphs. SS graphs are constructed using semi-direct products of a product of finite fields and Z2, where each such group yields a range of graphs of different degrees. SS graphs are the largest known Cayley graphs for degree larger than 30 and diameter 2, with order up to about 50% of the Moore bound.

Marcel Abas has found a general construction for Cayley graphs of any degree with order half of the Moore bound using direct product of dihedral groups Dm with cyclic groups Zn. It has been shown that, in asymptotic sense, the most of record Cayley graphs of diameter two is obtained by Abas construction. Using semidirect product of Zn&times;Zn with Z2 he has found (for degrees 13 &le; d &le; 57) largest known Cayley graphs in 34 cases of total 45 degrees and he constructed Cayley graphs of diameter two and of order of 0.684 of the Moore bound for every degree d &ge; 360 756.

A range of Cayley graphs of diameter 2 and degree larger than 12 was found by Eyal Loz using semi-direct products of cyclic groups.