Graphs

Codes and Combinatorial Structures

3.3 Graphs

Lemma 3.2.9 (10] Let V be a t-design with t

>

2. Then the group of automorphisms of the design acts faithfully on B.

The existence of multiply transitive groups may be used in order to construct designs, as in the following theorem.

Theorem 3.2.10 Let G be a t-transitive permutation group on a finite set

n,

with t

2

2, and suppose that

l:1

is a subset of

n,

with

1l:11 =

k,

1nl =

v, and 1< k < v - 1. Then the set B

=

{l:1⁹

I

g E G} is the set of blocks of a t-design V, and G is a group of automorphisms acting transitively on B.

Proof: See [10, Theorem 3.4.3] . •

Only a few symmetric designs are known to enjoy the property that a primitive nonsolvable group of automorphisms acts on points and blocks. In [57], Kantor classified all designs with 2-transitive group of automorphisms. In [38, Section 1], Dempwolff determined the symmetric designs V which admit G ::; Aut(V) such that G has a socle and is a primitive rank-3 group on points and on blocks.

In this thesis we shall be concerned mostly with self-dual symmetric l-(v, k, k) designs. In Theorem 5.2.1 we give a method to construct such designs. These designs will result from the primitive permutation representations of groups. In particular we are concerned with the primitive permutation representations of finite simple groups.

general relations on a set. The generality usually means that either the questions asked are too particular, or the results obtained are not powerful enough, to have useful consequences for design theory. There are instances where the two theories have interacted fruitfully. The unifying theme is provided by a class of graphs called strongly regular graphs, whose definition reflects the symmetry inherent in t-designs.

Definition 3.3.1 A graph

r =

(V, E), consists of a finite set of vertices V together with a set of edges E, where an edge is a subset of the vertex set of cardinality 2.

Our graphs are undirected (edges are not allowed to be ordered pairs), and without loops (two vertices comprising an edge are not equal) or multiple edges (a given pair of vertices can comprise at most one edge).

The complement of a graph

r

is the graph

l'

whose edge set is the complement of the edge set of

r

(relative to the set of all 2-element subsets of the vertex set).

Ifx is a vertex for a graph

r,

the valency of x is the number of edges containing x. If all vertices have the same valency, the graph is called regular, and the common valency is the valency of the graph. Thus an arbitrary graph is a 0- design, with block size k

=

2. A regular graph is a I-design.

Definition 3.3.2 A strongly regular graph with parameters (n, k,

>.,

^f-L) is a graph

r

with n vertices, not complete or null, in which the number of common neighbours of x and y is k, A or f-L according as x and y are equal, adjacent or non-adjacent respectively.

Remark 3.3.3 Notice that the complement of a strongly regular graph is strongly regular.

Definition 3.3.4 Let

r

be a graph with vertex set {X1,X2,'" ,xn }. The adjacency matrix A(r) = (aij) of

r

is the n x n matrix given by

if Xi and Xj are adjacent, otherwise.

Let r

=

(V, E) be a graph, and G be a permutation group on V. We say that G acts on r if, for all (a,(3) E E and g E G, we have (a9 ,(39) E E; that is, G is a group of automorphisms of r. The automorphism group Aut(r) of the graph r is the subgroup of Sv consisting of all automorphisms of r. We say that r is vertex-transitive if Aut(r) is transitive on the vertex-set V, and we say that r is a rank-r graph if Aut(r) is a transitive group of rank r on V.

The line graph of a graph r

=

(V, E) is the graph L(r)

=

(E, V) where e and

f

are adjacent in L(r) if e and

f

share a vertex in r. The complete graph Kn on n vertices has for E the set of all 2-subsets ofV and the null graph is a graph that has no edges at all. The automorphism group of the complete graph K n is the symmetric group Sn, since in this case any permutation of the vertices preserves adjacency.

The line graph ofK_n is the triangular graphT(n), and it is strongly regular with parameters C(~-l),2(n - 2), n - 2,4). The automorphism group of the triangular graph T(

n)

for

n >

4 is the symmetric group Sn' This follows by a Theorem of Whitney [94], which states that if r is a connected graph with more than 4 vertices, then Aut(L(r))

=

Aut(r). Now T(n)

=

L(K_{n )} implies Aut(T(n))

=

Sn for all n

>

4.

Let G be a rank-3 group of even order and let 01 , and O₂ be two orbitals other than the diagonal. Then G contains an involution ^T. Some pair x, y of distinct points are interchanged by an element ofG. Suppose that (x,y) E 0_{1 ,} then every pair in 01 is interchanged by an element ofG. So we can take the set of unordered pairs {x,

y}

for which (x,

y)

E 01 as the edge of a graph r on V. The fact that

0₁ and O₂ are orbitals implies that the number of common neighbours of two adjacent vertices, or two non-adjacent vertices, is constant; and the transitivity ofG shows that

r

is regular. So

r

is a rank-3 strongly regular graph.

Sporadic simple groups are often related with strongly regular graphs. For example, there is a strongly regular graph with parameters (162,105,81) and the MacLaughlin group of order 898, 128, 000 is a subgroup of index 2 of the automorphism group of this graph. Similarly there is a strongly regular graph with parameters (416,100,96) and the Suzuki group of order 448, 345, 497, 600 is a subgroup of index 2 of the automorphism group of this graph. For a survey on strongly regular graphs and rank-3 groups the reader is encouraged to consult [14, 15, 10].

The code formed by the span of the adjacency matrix of a graph

r

is also the code of the l-(v, k, k) design obtained by taking the rows of the adjacency matrix as the incidence vectors of the blocks; the automorphism group of this design will contain the automorphism group of the graph. Thus a relation is established between graphs, designs, codes and groups. The interplay between these structures will be more clear in Chapter 4, where these relations undergo a considerable development and are explicitly defined. In Chapters 7, 8 and 9 we construct codes from the adjacency matrices of the graphs defined therein and an interplay between codes and graphs is established.