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DA-ARO-D-31-124~G910 and the u.s. Air Force
Office of Scientific ReiJearoch under Contract AFOSR-68-1406 •
Durham" under Grant No.
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COMBINATORIAL
MATHEMATICS
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STEINER TRIEDERS AN)) STRONGLY REGULAR GP.APHS
by
Elisabeth Carriere
Department: of Statistics
and
FaauZte deB ScienaeB
University oj" North Carolina
Universite de Paris
" University of North Carolilla at Chapel Hill
Institute of Statistics rIimeo Series No. 600.26
}Iay
1970
STEINER TRIEDERS AND STRONGLY REGULAR GRAPHS
by
..
ELISABETH CARRIERE
Depaz-tment of Statistics
University of North Cazoolina at Chapel, Hil,l
and
Faaulte des Sciences
Universite de Parois
1. INTRJIlJCTIQ\J
A finite, undirected linear graph
regular of valence
exactly
1
Pu
n ,
l
G is strongly regular if it is
each adjacent pair of vertices is adjacent to
other vertices, and each non-adjacent pair of vertices is
adjacent to exactly
P~l other vertices.
We shall consider the Steiner configuration of 45 triangles formed by
27 straight lines contained in a general cubic surface.
The graph assOo-· .
dated with this configuration will be proved to be strongly regular.
The Steiner configuration has an interesting property about its triangles, and we shall see under which conditions graphs satisfying this
property are strongly regular.
2. THE STEINER CDiFIGUPATlm
The Steiner configuration, determined by the 27 straight Unes of any
cubic surface has the following properties (see [2] and [4]):
This l'eseaPch lJQB parrtial'Ly supported by the Army Reseazoch Office~
lAJ,phan.. unde1' Grant No. DA-AROD-31-124- G9IO and the u.S. Ail' F01'ce
Office of Scientific Research uruier Contract AFOSR-68-1406.
2
P.2.l.
Each straight line A meets ten others.
P.2.2.
The ten lines which meet any line A cut in pairs to form five
triangles with A.
P.2.3.
AlBlC l , A2B2C2 with no side in common,
there exists a unique triangle ~B3C3' such that Al A2A3 ,
Given two triangles
Bl B2B3 , Cl C2C are three more triangles, and
3
~B3C3
has no
side in common with AlBlC l and A2B2C2 •
These nine lines forming six triangles are called the Steiner trieder.
Let us consider the graph G associated with the Steiner configuration and defined as follows:
1.
The vertices of the graph G are the lines of the configuration.
2.
Two vertices of the graph G are adjacent if their corresponding
lines intersect in the configuration.
Thus each triangle in the configuration determines a triangle in the graph
(the sides of a triangle become its vertices).
Let us rewrite the properties
P.2.l, P.2.2, P.2.3
P.l:
Each vertex is adjacent to ten others.
P.2:
The ten vertices adjacent to any vertex x
for the graph G:
of the graph are
associated in adjacent pairs to form five triangles with x.
P.3:
Given two triangles
xlylzl
and x2YZzZ
of G with no vertex
in common, there exists a unique triangle x3Y3z3
such that
x l x 2x 3 , YlY2Y3' zlz2z3 arc three nore triangles and x3Y3zj. has
no vertex
i~
common with xlylzl
and x2Y2z2'
These nine vertices will also be said to form a Steiner trieder.
The graph G has 27 vertices and is regular of valance 10 by P.l.
3
PImEITla~
1.
The graph
G associated with the Steiner configuration is
strongly regular with the parameters:
1
v · 27,
n
• 10,
l
2
Pll • 1,
Pll· 5.
PIUf:
First, let us notice that the property
P. 3
implies obviously the
following property:
P.2.3
Given two triangles
xlylzl
and
x2Y2z2
of the graph
G,
with no vertex in canmon, each vertex of one triangle is
adjacent at least to one vertex of the other tr1aDgle.
Given any pair
a vertex
to
x,y
Z
and
x,y
of adjacent vertices of
such that
z.
xyz
is a triangle.
adjacent neither to
x
G.
vertex of the triangle
nor to
uvw
y,
xyz.
nor to
u
u
which is
and a triangle
z
z, the 27
uvw
which
P.2.3, each
is adjacent neither to
xyz,
x
nor
z.
x,y
is the unique vertex which is adj acent to
Thus we have
x
and
has to be adjacent to one vertex of
Then, the 24 vertices adjacent to
form with
x,y
But, by the property
which contradicts our hypothesis that
nor to
There are 24 vertices adjacent
If not, there exists a vertex
has no vertex in common with
y
there exists, by P.2,
If they -are distinct, we have, with
vertices of the graph
to
G,
1
Pll. 1.
x
and
z
and
are distinct and
z
y.
The ten vertices adjacent to any vertex
five triangles, two of them having only
x
x
as a common
vertex.
Any pair
triangles.
x, y
of nonadjacent vertices of
G dcu.eminetO- distinct
4
Let
xxIx
and
2
yy lY2
be two of these triangles.
two vertices in common, since
say
Xl· Y1'
P~l· 1. If they have one vertex in common,
and if the third vertex
have a triangle
l 2
is nonadjacent to
y
x
2
is adjacent to
1
Pu· 1.
which is impossible since
yx x
and
Y2
is nonadjacent to
x
If they have no vertex in common, the property
that is that at least one of
is adjacent to
of
Y1'Y
x.
x l ,x2
But if both of
are adjacent to
2
x,
They cannot have
is adjacent to
x ,x
then
l
2
yx x
l 2
we will
Hence,
x
2
for the same reason.
P. 2.3
y
XYlY2
must hold,
and one of
are adjacent to
or
y,
y,
Yl' Y2
or if both
will be a triangle
which contradicts that
P~l. 1. Thus each of the five triangles deter-
mined by the vertex
has one and only one vertex which is adj acent to
x
That is there are exactly five vertices adjacent to
have
x
and
y
and we
2
Pll. S•.
3. GPJfHS SATISfYING 1lE PRH:RtY P.3.
Consider a graph
with at least two triangles with no vertex in
G,
cammon and satisfying P.3.
Xi' Yi' zi'
i
D
1 to 3
G has at least nine distinct vertices
which form a Steiner tr1eder
TO'
that is,
which satisfy the following:
P.3.3
Given two triangles which are two rows of
Xl Y1 zl
~ Y2 z2
x 3 Y3 z3
the third row is the unique triangle such that each column is
a triangle
then we have the following property:
P.3.4
for
i . j,
i, j • 1 to 3.
y.
5
Yi
and Y
j
and Zj
are nonadjacent
z1
and x
are nonadj acent
Xi
j
are nonadjacent
PfmF:
Suppose xl
adjacent to Y2 ,
then xl x2Y2 and x l Yl Y2 will be
two triangles with no vertex in common with
be adjacent to xl'
If
have
Xl
x2
x3
z3
zlz2z3.
Then
z3 has to
or to
Y , or to x and Y •
2
2
l
is adjacent to xl' then z l xl z 3 is a triangle and we
Yl zl
Y2 xl
Y3 z3
where each row and each column 1s a triangle which contradicts
If
have
1s adjacent to
P.3.3
is a tr1angle and we
Y2'
Xl Yl Y2
x 2 Y2 z2
x3 Y3 z3
where each row and each column is a triangle which contradicts P.3.3
If
is adjacent to Yl
are two triangles and we have
and x2 '
xl Yl zl
where each row and each column is a triangle which contradicts P. 3.3
6
Thus
xl
is not adjacent to Y2
other pair and 1'.3.4 holds.
This implies three more properties:
Any triangle xyz
1'.3.5
and we can reason similarly for any
of
G which is not contained in the trieder
TO has at most one vertex in CODlllon with it.
triangle xyz
p.3~6
.Any
1'.3.7
For any pair xy
one vertex
If xyz
of adjacent vertices of G,
z adj acent to x
is a triangle of
vertices xy
of G is contained in a Steiner trieder.
G
and y.
not contained in TO'
in common with TO'
say x· xl'
distinct from Y since they are not adjacent to
then
z;e zl and xyz· xlYlz
that is
which has two
then Y2z2Y3z3
Xi
Suppose
are
y. Y1'
has no vertex in common with x2Y2z2
x3y3 z3 and 1'.2.3 implies that
are adjacent to z,
there is at most
z2
zZ2z3
and z3
and
nonadjacent to Xl
is a triangle.
and Yl
We have Xl Yl z
x2 Y2 z2
x 3 Y3 z3
Where each row and each column is a triangle which contradicts 1'.3.3
since
z;e zl.
Thus 1'.3.5 is satisfied.
It follows that any pair of adjacent vertices in TO
unique triangle.
determine a
It also follows that for any triangle xyz
of
G there
is a triangle of TO which has no vertex in common with it and by 1'.3
they must form with a third triangle a Steiner trieder.
Thus
P.3.6 and
1'.3.7 are satisfied.
Let us introduce now one more property:
1'.3.8
Each vertex of the graph G is contained in at least one
triangle.
7
A graph
G with at least two triangles with no vertex in common,
satisfying P.3.8 and P.3 also satisfies:
P.3.9
Each vertex of G is contained in a Steiner trieder.
THEOIm 1.
A finite. undirected linear graph
G,
containing at least two
triangles with no vertex in common satisfies P.3.9· iff it is
strongly regular.
v
lID
3(2r-l),
n
l
Its parameters are:
lID
1
2r,
r
Pll ... 1,
~
2
PfmF:
Let
xy
be a pair of adjacent vertices of
P.3.8 imply that there is a unique vertex
z
then P. 3. 7 and
G,
and
which is adjacent to x
1
The valence n of each vertex x of G is even and
x
nx
x is contained in 2
triangles. Let xy be a pair of nonadjacent
y.
Then Pll'" 1.
vertices in G,
and xxl x 2 ' "lY2 be two triangles. If they have one
vertex in common, Xl III y1' and if x 2 is adjacent to y, or Y2
adjacent to x,
diets that
then yx x2 or XYIY2 will be a triangle which contral
1
PIl'" 1. If they have no vertex in common they must form
with a third triangle a Steiner trieder and P.3.4 implies, since x
is
not adjacent to Y that one and only one of x ,x2 is adjacent to y,
l
and one and only one of
Yl'Y2
is
adjacent to x.
of triangles containing respectively x and
n
n
in x triangles and Y in .J.. triangles.
2
2
n
-!X
there are
vertices of G adjacent to
2
Now let x be a vertex of valence n ==
l
> 2.
y.
And this for any pair
But
x
is contained
That implies nx ... ny c:
xy.
2r
in
G,
r
must be
n
xy
and
8
xyz
For any vertex
y
is a triangle.
Then, for every
adj acent to
x
nor to
adjacent or not to
is regular.
x
adjacent to
y
and then
x.
there is a vertex
zi
n
n
D
y
has the same valence
1:1
zi
G is
3(2r-l).
A strongly regular graph
G,
3(2r-l),
n
1
1:1
2r,
l
= 2r.
is not
Thus any vertex
.. 2r, and the graph G
2
nl
Pl1""2" .. r. Then the
The converse is obvious.
with at least two triangles with
no vertex in common satisfies
a
n
z,
such that
~
It also is strongly regular with
number of vertices in
V
adjacent to
z
P.3 iff its parameters are
1
Pl1 .. 1,
2
Pu
1:1
r,
r > 2.
Strongly regular graphs with such parameters have been proved by
Seidel in
[3]
to exist only for
r
1:1
1, 2, 3, andS.
I would like to express my thanks to
suggested to me the study of the property
I. M. Chakravarti
P.3.
see
[1].
who
REFERENCES
[1]
Chakravarti, I. M., Some properties and applications of Hermitian
varieties in a finite p~ojective space PG(N,q2) in the
construction of strongly regular graphs (two-class association
schemes) and block designs.
Institute of Statistics Mimeo
Series No. 600.23. March 1970, U.N.C., Chapel Hill.
[2]
J"ordan, -C., Trait~ des SUbstitutions et des Equaticns
Gauthiers-Vi11ars, Paris (1870).
[3]
Seidel, J. J., Strongly regular graphs with (-1, 1, 0) adjacency
matrix having eigenvalue 3. Linear AZgebraa and Its AppUaations,
1 (1968), 281-298•
[4]
Steiner, J., Uber e1ne besondere curve dritter Klasse (und vierten.
Grao1.es),. J"ou.:ma'L fUra die Reine una. Angewandte Mathematik~
Berlin, 1857, Vol. 53, pp. 231-237.
..
A19~brique8.