Scheme with d classes if
1. A0= I.
2. A1+ A1+ ⋯ + Ad = J the all-one matrix.
3. For all i there is a j such that ATi = Aj.
4. AiAj= ∑kpki , jAkfor nonnegative integers pki , j. The pki , jare called the iter-section numbers, parameters or structure constants of the scheme.
Example: Let G be a transitive permutation group on Ω = {1, . . . n}. Then G acts on Ω × Ω, the orbits being called orbitals. For each orbit we define a 0/1 matrix A = (ax , y) such that ax , y= 1 if and only if (x, y) is in the orbit.
As one orbit consists of pairs (i, i) we may assume that A0 = I, and clearly
∑ Ai= J. Furthermore, is (i, j) and (x, y) are in the same orbit, clearly also (j, i) and (y, x) are in the same orbit, shoing that ATi = Aj.
Finally, let Ai = (ax , y) and Aj = (bx , y). Then the (x, y) entry of AiAj is
∑zax ,zbz , y, that is the number of z’s such that (x, z) is in the first orbit and (z, y) is in the second orbit, thus it is a nonnegative integer. Call this number qi , j(x, y).
Furthermore, if (x′, y′) lies in the same orbit as (x, y), that is for g ∈ G we have that xg = x′and yg = y′, we have that the z′such that (x′, z′) is in the first orbit and (z′, y′) in the second are exactly the elements of the form zg. Thus the values of qi , j(x, y) are constant on the orbit of (x, y), that is we can set pki , j= qi , j(x, y) for an arbitary (x, y) in the k-th orbit.
X.3 Moore Graphs
Definition X.17: The diameter of a graph is the maximum distance between two vertices.
The girth of a graph is the length of the shortest closed path.
A connected graph with diameter d and girth 2d + 1 is called a Moore graph.
Example: The 5-cycle and the Petersen graph both are both examples of Moore graphs.
Lemma X.18: Let Γ be a Moore graph. Then Γ is regular.
Proof: Let v, w be two vertices at distance d and P a path connecting them. Consider a neighbor u of v that is not on P. Then v must have distance d from w, as otherwise w − u − v − w would be a cycle of length < 2d + 1. Then there is a unique path from u to w (as otherwise the differing paths would give a cycle), it will go through one neighbor of w. Different choices of u yield (again cycle length) different neighbors of w and thus w has at least as many neighbors as v. By symmetry thus v and w must have the same degree. Since we have seen that all neighbors of v, that are not
on P also have distance d from w, by the same argument they also have the same degree, as will be all neighbors of w not on P.
Now Let C be a cycle of length 2d+! involving a vertex v. By stepping in steps of length d along the circle the above argument shows that all vertices on C must have the same degree. If x is a vertex not on C take a path of length i from x to C.
By taking d − i steps along C we have that x is at distance d from a vertex on C and
thus has the same degree. ◻
Lemma X.19: A Moore graph Γ of diameter d and vertex degree k has 1 + k((k − 1)d− 1)/(k − 2) vertices.
Proof: Vertex degree k implies (easy induction) that there are at most k(k − 1)i−1 vertices as distance i ≥ 1 from a chosen vertex v.
Suppose that w is at distance i. Then there is one neighbor of w art distance i − 1, and no other at distance i (as otherwise we could form a cycle using w and this neighbor). Thus k − 1 neighbors of w must be at distance i + 1 from v, which shows that there are at least that many neighbors.
Thus there are in total
1 + k + k(k − 1) + k(k − 1)2+ ⋯ + k(k − 1)d−1= 1 + k(k − 1)d− 1 k − 2
vertices. ◻
Consider a Moore graph of diameter 2 Then G has 1 + k + k(k − 1) = k2+ 1 vertices and girth 5. Thus two adjacent vertices have no common neighbor, and two non-adjacent vertices have exactkly one common neighbor.
Thus Γ is strongly regular and its adjacency matrix A satisfies A2= kI + (J − I − A),
and A has eigenvalues k and
λ±= −1 +√ 4k − 3
2 .
If 4k − 3 is not a square, then λ±will be irrational and thus the multiplicities m±
must be equal (and thus be (n − 1)/2 = k2/2.
As A has only zeroes on the diagonal, it has trace 0. But the trace is the sum of all eigenvalues. Thus (with the irrationalities cancelling out)
0 = k +k2 2 (−1
2 +−1 2)
and thus k = 2. A graph of valency 2 must be a cycle, thus the only possibility here is the 5-cycle.
X.3. MOORE GRAPHS 175 If 4k − 3 is a square, it must be the square of an odd number, 4k − 3 = (2s + 1)2. We solve for k = s2+ s + 1 and the eigenvalues of A are k, s, and −s − 1, the latter two with multiplicities f and g such that f + g = n − 1 = k2. The trace of A is k + f s + g(−s − 1) = 0. These are two linear equations in f and g, which gives
f = s(s2+ s + 1)(s2+ 2s + 2)
2s + 1 .
This number must be an integer. We expand the numerator as
s5+ 3s4+ 5s3+ 4s2+ 2s = 1
32((16s4+ 40s3+ 60s2+ 34s + 15)(2s + 1) − 15) . We conclude that 2s + 1 must divide 15, which implies that s = 0, 1, 3, 7. This corre-sponds to k = 1, 3, 7, 57, respectively n = 2, 10, 50, 3250.
The choice of n = 2 vertices is spurious, thus a Moore graph of diameter 2 and valency k can exist only for k = 2, 3, 7, 57 and associated n = 2, 10, 50, 3250.
We already have seen the 5-cycle for k = 2. For k = 3 the Petersen graph is an example. There is also an example for k = 7, the Hoffman-Singleton graph. The case k = 57, n = 3250 is open.
It has been shows that these three, possibly four, cases are the only Moore graphs of diameter 2 and that for diameter d ≥ 3 the 2d + 1 cycle is the only Moore graph.
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Index
diameter, 173
Elementary Symmetric Function em, encode, 14678
181
primitive n-th root of unity, 158 Principle of Inclusion and Exclusion,
Stirling number of the second kind, strongly regular, 16810
transitive, 98, 167 two-line arrays, 86 value of a flow, 59 vertex cover, 52 vertex path, 57 vertex separating, 57 weight, 149
weight enumerator, 153 Witt design, 163 words, 147
wreath product, 105 Young diagram, 67 zeta function, 46
Some Counting Sequences
Bell OEIS A000110, page 27
B0=1, B1=1, B2=2, B3=5, B4=15, B5=52, B6=203, B7=877, B8=4140, B9=21147, B10=115975
Catalan OEIS A000108, page 22
C0=1, C1=1, C2=2, C3=5, C4=14, C5=42, C6=132, C7=429, C8=1430, C9=4862, C10=16796
Derangements OEIS A000166, page 24
d(0)=1, d(1)=0, d(2)=1, d(3)=2, d(4)=9, d(5)=44, d(6)=265, d(7)=1854, d(8)=14833
Involutions OEIS A000085, page 31
s(0)=1, s(1)=1, s(2)=2, s(3)=4, s(4)=10, s(5)=26, s(6)=76, s(7)=232, s(8)=764 Fibonacci OEIS A000045, page 16
F0=1, F1=1, F2=2, F3=3, F4=5, F5=8, F6=13, F7=21, F8=34, F9=55, F10=89, F11=144 Partitions OEIS A000041, page 67
p(0)=1, p(1)=1, p(2)=2, p(3)=3, p(4)=5, p(5)=7, p(6)=11, p(7)=15, p(8)=22, p(9)=30
183