Research supported by National Science Foundation Grant No. GP-8624
and by United States Air Force Grant No. AFOSR 68-1415.
A GENERALIZATION OF MOORE GRAPHS OF DIAMETER TWO
by
T.A. Dowling
and
Department of Statistics
University of North Carolina at Chapel Hill
R.C. Bose
Institute of Statistics Mimeo Series No. 600.11
JULY 1969
A Generalization of Moore Graphs of Diameter Two
by
R.C. Bose
and
T.A. Dowling
Department of Statistics
University of North Carolina at Chapel Hill
1.
INTRODUCTION.
A finite, undirected linear graph
regular [1] if it is regular of valence
tices is adjacent to exactly
n ,
l
G is strongly
each adjacent pair of ver-
other vertices, and each non-adjacent
pair of vertices is adjacent to exactly
other vertices.
The no-
tation is that of two-class association schemes, to which strongly regular graphs are abstractly equivalent.
As defined by Bose and Shimamoto
[3], a two-class association scheme consists of a finite set of
v
ele-
ments together with a scheme of relations among them such that
(i)
any two distinct elements are either first associates or
second associates;
(ii)
(iii)
each element has exactly
n.
1
i-th associates
if two elements are i-th associates, the number
(i
=
i
Pjk
1,2);
of
elements which are j-th associates of the first and k-th
associates of the second is independent of the original
pair, and
(i, j, k
= 1,
2).
If the elements of a set on which is defined a two-class association
scheme are taken as the vertices of a graph
cent (non-adjacent) in
G,
with two vertices adja-
G iff the corresponding elements are first
Research supported by National Science Foundation Grant No. GP-8624
and by United States Air Force Grant No. AFOSR 68-1415.
2
(second) associates, it follows from the above definition that
strongly regular.
G is
Conversely, given a (finite) strongly regular graph
G with parameters
1
2
a two-class association scheme is
n l , Pll' Pll'
obtained by associating the elements with vertices and defining two elements to be first (second) associates iff the corresponding vertices are
adjacent (non-adjacent).
This latter equivalence is a result of the
fact that the constancy of the three parameters
ficient to guarantee the constancy of the remaining parameters, and to
determine them uniquely through well-known relations.
of the eleven parameters
i
v, n., P'k
1
J
(i, j, k
= 1, 2)
Thus only three
are independent.
The generalization of the above definition to an m-class association
scheme is straightforward, but for
tation is usually lost.
m2 3
the graph-theoretic represen-
Certain classes of graphs, however, do corre-
spond to m-class association schemes in which the association class to
which two elements belong is determined by the graph-theoretic distance
between the corresponding vertices.
regular of diameter
m.
Such a graph is said to be metrically
A connected strongly regular graph
2
(Pl1
>
0)
is therefore metrically regular of diameter two.
The present paper considers two questions associated with strongly
regular graphs.
In Section 3, we investigate the consequences of
dropping the regularity condition in the definition of a strongly regular graph.
We show that the irregular graphs satisfying the remaining
two conditions fall into two infinite classes and have a very simple
structure.
In Sections 4 and 5, we consider the class of strongly regular
graphs with
2
Pll
= 1.
This investigation is motivated by the proof of
Theorem 1 in Section 3, where some interesting geometric properties of
3
these graphs are revealed.
A characterization in terms of properties of
the maximal cliques is given (Theorem 2), which shows that strongly reg2
PII = 1 provide a generalization of Moore graphs of
ular graphs with
diameter two [6].
The main results of Section 5 (Theorems 3 and 4) establish necessary
conditions for existence of strongly regular graphs with a given value
A consequence of Theorem 4 is that the class
and with
of
of all such graphs is finite if
values for
I
A table of possible parameter
PII # 3.
is given.
In Section 6, we define the Moore geometries
MG(r, k, d),
the
graphs associated with which may be regarded as generalizations of Moore
graphs of diameter
d.
However, the investigation of the detailed
properties of these geometries and graphs is reserved for a future
communication.
2.
NOTATION AND TERMINOLOGY.
We consider finite, undirected
linear graphs.
A graph will be denoted
vertex set and
E
subsets of
We use the symbol
V.
the edge set.
element subset of
according as
For
x, y
V.
(x, y)
V,
E
E
or
A(x,y)
of
A(x).
x
~(x,y)
E
(V, E),
(x, y)
(x, y
E
where
V is the
are two-element
to denote an arbitrary two -
x, yare adjacent or non-adjacent
~
E.
we define
A(x)
The valence of
=
The elements of
Two vertices
E
G
=
{z:
=
A(x) n A(y) .
V will be denoted
(x,z)
E
d(x),
denotes the cardinality of
E}
equal to the cardinality
A(x,y).
4
A graph
G is regular if all vertices have equal valence and com-
plete if any two vertices are adjacent.
any two of which are adjacent.
A clique is a set of vertices,
A clique is maximal if it is not properly
contained in some other clique.
3.
IRREGULAR GRAPHS WITH
CONSTANT.
notation, a strongly regular graph is a graph
1
fixed non-negative integers
(AI)
(A2)
(A3)
2
n l , Pll' Pll'
d(x) = n '
l
1
L'>(x,y)
PH'
2
L'>(x,y) = Pll'
G
=
In terms of our
(V, E)
such that for
the following hold:
x
E:
V,
(x,y)
E:
E,
(x ,y)
~
E.
We may assume additionally, to exclude trivialities, that strongly regular graphs considered are not complete.
graph with parameters
Given a strongly regular
elementary counting arguments es-
tablish the relations
=
(3.1)
is the number of vertices non-adjacent to any given vertex,
where
and
v
Let
is the total number of vertices.
r
~
1,
k
follows:
GO(r,k,s)
ponents,
r
s
~
2,
has
s
~
1,
rk + s
and define a graph
vertices and
of which are complete graphs with
of which are isolated vertices.
r + s
k
GO(r, k, s)
as
connected com-
vertices each, and
Figure 1 illustrates
G (2,4,1).
O
5
o
Figure 1.
Evidently
but is not regular.
lows:
G (r,k)
l
r
has
complete graphs with
tex in common.
2
pU = k-2, pU
0,
We define
=
Next suppose
1
satisfies (A2) and (A3) with
GO(r,k,s)
~
2,
r
k
~
r(k-l) + 1
k
2,
~
1,
s
~
l}.
and define a graph
Gl(r,k)
vertices and consists of
r
as foledge-disjoint
vertices each, all of which have a single ver-
Figure 2 illustrates
Figure 2.
G (3, 4).
l
G (3, 4).
1
6
Note that
1
Pll
Gl(r,k)
2
= k-2,
is not regular but satisfies (A2) and (A3) with
= 1.
Pll
Finally, let
G (k)
l
{Gl(r,k):
=
r ~ 2}.
We then have
THEOREM 1.
Let
P
1
2
II , PII be non-negative integers and let
G
= (V,E)
be a graph such that
(A2)
l',(x,y)
(x,y)
E
E,
(A3)
t>(x,y)
(x,y)
~
E.
Then exactly one of the following holds:
(a)
G is regular; hence strongly regular,
2
Pll
2
Pn
(b)
(c)
PROOF.
Let
x
0
and
G
E
GO(P ll + 2) ,
=
1
and
G
E
Gl(P ll + 2) .
V,
E
ordered triples
Yl' Y2
A(x),
E
1
=
and suppose
(y l' Y2' z)
Z
E
1
= n.
d(x)
Consider the number
V - x,
of distinct vertices of
If
A(Yl' YZ)'
Z
E
A(x),
1
1
1) ways, while if
PU(P II
2
2
Pll (Pu - 1) ways. Letting v
the pair
~
A(x) ,
T of
such that
(Y l ' Y2)
can
be chosen in
Z
chosen in
be the number of vertices
in
(Yl' Y2)
can be
we have
G,
(3.2)
is fixed, the ordered pair
Alternatively, if
in
with
1
1
Pll (Pn- I )
Y2
~
ways wi th
A(Yl)'
Thus
YZ
E
A(y l )
and in
(Y
I
z'
can be chosen
z)
2
(n-Pll-l) (Pll-l)
ways
7
(3.3)
T
Equating (3.2) and (3.3) and simplifying yields
0.
(3.4)
If
2
in this case
If
(3.4) has a unique non-negative solution for
Pll > 1,
2
Pu
1
Pu + l.
n, so
is regular.
G
= 0,
both solutions of 0.4) are non-negative:
2
PH
Since
=
every connected component of
0,
G
n
= 0,
must be
complete, so that the only possible components are isolated vertices
1
Pll+2
and complete graphs with
G
types occur, so
vertices.
If
G
is not regular, both
1
G (Pll+2).
O
E
Before proceeding further, we shall prove the following lemma:
LEMMA 1.
If
G is a graph satisfying the conditions (AZ) and (A3) of
Z
PH = 1,
Theorem l, wi th
Let
(x,y)
E
E
then for any
non-adjacent vertices
and define
x u Y u A(x,y).
(3.5)
K(x,y)
If
is not a clique, there exist vertices
K(x,y)
such that
Thus
(zl' zZ)
K(x,y)
(PI)
~
E,
is a clique, and clearly maximal.
Every pair
Z
PH
=
1
>
zl' z2
which contradicts (A3) , since
x,y
0,
clique has the form (3.5).
K(x,y)
E
x,y
A(x,y)
E
A(zl,z2).
We have
of adjacent vertices in
in a unique maximal clique
Since
and
G is contained
defined by (3.5).
G has no isolated vertices, so every maximal
Thus
8
(P2)
If
Z
x, y, z
A(x,y),
E
1
Pll+2
Every maximal clique contains exactly
so
(P3)
are the vertices of a 3-cycle in
z
E
G,
vertices.
then
K(x,y).
The vertices of any 3-cycle in
are contained in a max-
G
imal clique.
Suppose
order.
w,y
w, x, y, z
Since
K(x,z)
E
(P4)
w,y
E
are the vertices of a 4-cycle in
A(x,z),
(A3) implies that
(x, z)
E,
E
in cyclic
G
so
and we have
The vertices of any 4-cycle in
G
are contained in a max-
imal clique.
Let
KO
xo' Yo
= K(xO'ZO),
be non-adjacent vertices of
LO
= L(yO'ZO)'
Then
is trivial unless one of xa,yo
cliques.
o
E
K1 - xo·
Yl
= A(xl,y)·
By (PI),
and
y
xl tKO'
K
l
LO'
E
containing
and by (P4)
Then (P3) and (P4) imply
Yl
~
Since
are non-adjacent by (P4).
(x1,y)
L '
O
~
E.
A(Yd - LO such that
(x1'Yl)
K and
O
d(x )
O
that
Since
= d(yO)'
so there is a second
Yl F zo'
and
Yl
E
A(yO)'
Thus
A(x ) - K
O
O
correspond iff
LO have equal cardinality, it follows
This completes the proof of Lemma 1.
To complete the proof of Theorem 1,
graph
and let
Let
Hence
and the vertices of
E.
The result
xo'
there is a one-one correspondence between the vertices of
E
= A(xO'YO)'
Zo
is contained in at least two maximal
= K(Yl'YO) containing yO'
Ll
Let
KO and
E
Assume there is a second clique
xl
clique
X
G.
we have to show that if a
G, satisfying the conditions of Lemma 1, is irregular then
1
G E
Gl (Pll+2).
G.
Then
Let
d(x )
O
then by (P4),
and
Yo
be any two non-adjacent vertices of
say.
w
Let
Zo
= A(xO'YO)'
is non-adjacent to at least one of
If
and
w
E
V-z O'
9
Hence
d(w)
= nle
If
G is irregular,
adjacent to every vertex of
V-z '
O
Hence
is
otherwise Lemma 1 would be violated.
It follows from (PI) and (P2), that there exists a set of maximal cliques
of cardinality
of
v-z
1
Pll+2
with
zo
as a common vertex such that any vertex
belongs to one and only one of these cliques.
O
tices of
V-z
O
belonging to different cliques of the set must be non -
adjacent by (P3).
Hence
COROLLARY.
satisfies (A2) and (A3) with
If
Also two ver-
G
then
G is
strongly regular.
In analogy with Theorem 1, one might ask whether the class of all
graphs of diameter two satisfying (AI) and (A2) but not (A3) can be
similarly characterized, or the class satisfying (AI) and (A3) , but not
(A2).
Actually these two questions are equivalent, since
G has diam-
eter two and satisfies (AI) and (A2) iff the complementary graph
G
satisfies (AI) and (A3) , but not (A2) , with
-2
Pll
=v
1
- 2n l + Pll'
where
v
is the number of vertices in
G.
Sev-
eral examples of such graphs are known, but no complete characterization
is presently available.
4.
A CHARACTERIZATION OF STRONGLY REGULAR GRAPHS WITH
2
Pll
= 1.
Properties (Pl)-(P4) obtained in the proof of Theorem 1 for the case
lead to a characterization of strongly regular graphs with
in terms of properties of the maximal cliques.
We present the
result in this section (Theorem 2) along with a description of the connection of these graphs with Moore graphs of diameter two.
10
We consider, therefore, the class of all graphs satisfying (Al)-(A3)
2
Pu
with
= 1.
THEOREM 2.
2
Pll
A graph
G
=
(V, E)
is a strongly regular graph with
= 1 iff there exist integers r, k such that r
~
k
~
2
and the
following hold:
(Ql)
Every maximal clique contains
k
(Q2)
Each vertex is contained in
(Q3)
Each pair of adjacent vertices is contained in a unique max-
r
vertices;
maximal cliques;
imal clique;
(Q4)
Given two non-adjacent vertices
maximal clique containing
clique containing
The parameters of
n
n
=
l
taining
x, y
r(k-l) ,
=
k - 2,
=
l.
G satisfies (Ql)-(Q4) with
(Ql)-(Q3) imply
(x,y)
y.
2
r(r-l) (k-l) ,
2
1
For
= r(k-l)
d(x)
E
E,
for all
we denote by
guaranteed by (Q3).
L
l
Suppose there is a vertex
= K(y,z).
Since
z
~
K,
tinct, and the maximality of
x
K(x,y)
Then if
E
r
z
E
~
~
k
2.
Clearly
V.
the maximal clique con(x,y)
K = K(x,y)
other vertices of the maximal clique
A(x,y).
which intersects a maximal
= 1 + r(k-l) + r(r-l) (k-l)2.
Pll
2
Pll
Suppose
there is a unique
G are then given by
v
PROOF.
x
x, y,
E
the
k-2
are clearly in
A(x,y) - K.
the maximal cliques
E,
Let
K
l
K, L , K
l
l
= K(x,z),
are dis-
K implies the existence of a vertex
11
z.
K non-adjacent to
W E
taining
K , L
l
l
are two maximal cliques con-
which intersect the maximal clique
z
contradicts (Q4).
Suppose
taining
Then
x
~(x,y)
Hence
(x,y)
~
E.
Let
= k-2
for all
x
and
y
G
L
containing
y
Conversely, if
G
v
~(x,y)
~
r
2
if
Xl
E
K -x '
(x 'Yl)
1
E.
E
vertices
r
2
Pl1
= 1,
vertices
clique
L.
J
(1 ::; j
properties
G is regular, every
of maximal cliques.
r
~
k.
Since
Let
X
o
and let
Putting
z
Y1
Xl' xi
YO.
1
= y'1
is not
o=
be
Let
be
For each vertex
A(YO)-L O such that
E
in
K1-x O the corresponding
are in the same maximal clique
y
G
Ll , L2 , .•• , Lr - 1
maximal cliques containing
then we contradict (P3) if
k-l
It
follow from (3.1).
and there exists
If for two vertices
Y1' Yi
~ E.
(x,y)
P~l = 1.
there is a unique vertex
O
1
By
K.
Then (Q4) is an immediate consequence of
a second maximal clique containing
r-l
K
Thus there is
as in the proof of Lemma 1.
the remaining
If
1
= r(k-l), Pll
= k-2,
1
It remains to establish the inequality
complete,
1
In addition, since
(Ql)-(Q3) follow.
(Q3) and the condition
n
and
vertex is contained in the same number
y.
then by reversing the
is strongly regular with
(Pl)-(P4) of Section 3 hold.
which
E.
which intersects
is strongly regular with
The expressions for
1
= Pll+2,
y,
in (Q4) we obtain a contradiction.
has cardinality one, so
K n L
follows that
k
E
which intersects a maximal clique containing
a unique maximal clique
(Q3) ,
(x,y)
w,
K be the unique maximal clique con-
intersects two maximal cliques containing
roles of
containing
K
and (P4) i f
L.
J
(1::; j
Yl # yi·
::;
r-l) ,
Hence the
are distinct, and no two appear in the same maximal
::; r-l).
Thus
k-l ::; r-l,
and the proof is complete.
12
In vies of Theorem 2, we may denote an arbitrary strongly regular
graph with
1
by
G(r,k),
where
r
~
k
meanings attached by (Q2),(Ql) respectively.
~
2
and
r, k
have the
It is, of course, possible
that there may exist several non-isomorphic graphs satisfying (Ql)-(Q4)
for given values of
r, k,
so that the notation
G(r,k)
does not
uniquely define the graph.
Strongly regular graphs with
2
Pll
=1
are a generalization of
Moore graphs of diameter two [6].
A Moore graph
graph of valence
d
r
and diameter
M(r,d)
is a regular
for which the number
v
of ver-
tices attains the upper bound
1 + r + r(r-l) + ... + r(r-l)
imposed by these requirements.
G(r,2)
exists only for
and possibly
M(r,2),
is a Moore graph of diameter two, and the converse can
also be easily established.
G(r,2),
In particular, for
= 1 + r + r(r-l) .
v
Thus
d-l
r
= 57.
r
In [6], it is shown that
=2
(pentagon),
3
M(r,2),
hence
(Peterson graph),
That these are the only possibilities for
7,
r
will follow as a special case of Theorem 4 below.
Moore graphs of diameter two have girth five, hence no 3-cycles or
4-cycles.
A similar structure is exhibited by
G(r,k)
if we define
girth in terms of minimal cycles, no three vertices of which lie in the
same maximal clique.
5.
WITH
NECESSARY CONDITIONS FOR EXISTENCE OF STRONGLY REGULAR GRAPHS
2
Pll
= 1. While the existence of three Moore graphs of diameter two
13
(G(r,2)
with
for
k
2
>
r
= 2,
3, 7)
is known, no strongly regular graphs
are presently known.
conditions for the existence of
G(r,k)
In this section, we derive necessary
G(r,k).
Our first result is an easy
consequence of Theorem 2.
THEOREM 3.
A necessary condition for the existence of a graph
is that
divide
PROOF.
k
Let
b
r(r-1) 2.
be the number of maximal cliques in
counting vertex-maximal clique incidences, we have
divides
vr.
divides
G(r,k).
bk
= vr,
Then by
so
k
Since
1 + r(k-1) + r(r-1) (k-1)
v
k
G(r,k)
vr
iff
2
=
[r(r-1)k - r(2r-3)]k + (r_1)2,
k
divides
2
r(r-1) •
In deriving stronger necessary conditions for the existence of
G(r,k),
we shall make use of a theorem on two-class association schemes
due to Bose and Mesner [2].
We state the theorem here in its equivalent
form for strongly regular graphs.
THEOREM (Bose-Mesner).
eters
of
G.
Let
1
2
v, n , n 2 , P11' P11'
1
Then
G be a strongly regular graph with paramand let
A
be the adjacency matrix
is a simple characteristic root of
two other distinct characteristic roots
where
= A(G)
A,
and there are
8 , 8
with multiplicities
2
1
14
N
(n -n ) + y(n +n ),
1 2
1 2
1
2
1,
Pll - Pn
1
2
2n l - Pu - Pn - 1,
2
y + 2S + 1.
=
y
S =
l::.
=
The requirement that
a , a
l
2
ditions on the parameters of
G.
be integral imposes necessary conApplying the theorem to
G(r,k),
we
have, from Theorem 2,
N
2
r(k-1) [1 + (r-1) (k-3)]
=
y
k - 2,
S =
2rk - 2r - k,
l::.
(k-1)
(5.1)
If
=
r = k = 2.
The graph
G(2,2)
l::.
a
= 2b
2
k-1
= (k-l+a) 2 .
+ 4(r-l) (k-l)
and
k-1 + a
must be of the same parity, we put
r - 1
k-1
divides
k - 1
(5.4)
where
We assume
in (5.2) and simplify, obtaining
(5.3)
Thus
is, of course, the pentagon.
must be a square, so we set
(k-l)
Noting that
N must vanish, which implies
(r,k) # (2,2).
below, therefore, that
(502)
+ 4(r-l)(k-l).
is not a square, then clearly
l::.
Then
2
f
b
=
2
,
say
b
2
=
m(k-1).
Define integers
c, f
by
2
c f,
is square-free.
Then
b
2
=
2
c fm,
and since
f
is square
15
free,
2
m= t f
m has the form
for some integer
t.
Thus
b = cft,
so by (5.3),
=
r - 1
(5.5)
(t+c)tf.
Substituting (5.4) and (5.5) into the expression for
~
given by
(5.1), we obtain
Ii
(5.6)
cf (2t+c) .
=
Consider now the expression for
we may regard
N as a second degree polynomial in
(5.4) and (5.5) into
(k_l)2,
Pk(t)
k.
k
fixed,
Substituting
t
with
Factoring out the constant
we have
=
N
where
r.
For
N gives a fourth degree polynomial in
integral coefficients determined by
factor
given in (5.1).
N
is a fourth degree polynomial in
ficients determined by
k.
t
with integral coef-
Then by (5.6),
(5.7)
By the Bose-Mesner Theorem,
N/Ii
must be integral.
We consider
two cases.
CASE 1.
odd.
k
t
1 (mod 4):
In this case
We may therefore multiply
3
c fP (t)
k
c
is odd, so
2
4
is
by any power of two without
affecting its divisibility (or lack of divisibility) by
Choosing the factor
2t + c
2t + c.
(or any higher power) permits us to carry out
16
the polynomial division in (5.7) to obtain a quotient polynomial with
integral coefficients, plus a remainder term.
here.
We omit the calculations
The result is
=
where
Qk(t)
is a cubic polynomial in
t
with integral coefficients,
and
=
Hence, if
G(r,k)
CASE 2.
c(k-l)(k-5)(k
exists,
2t + c
k - 1 (mod 4):
2
- 4k - 1).
~.
must divide
In this case,
c
=
2d
is even, and (5.7)
becomes
The polynomial division yields
where
Qk(t)
is a cubic polynomial in
t
with integral coefficients,
and
2
d(k-l)(k-5)(k +3)/16.
Rk
Thus the existence of
G(r,k)
2t + c
equivalently, that
implies that
~
divides
=
t + d
2Rk'
divides
Rk'
or
Our results are sum-
marized in
THEOREM 4.
Let
k
~
2
and let
c, f
k - 1
=
be positive integers defined by
2
c f,
17
where
f
is square-free.
of a graph
G(r,k),
Then a necessary condition for the existence
other than
G(Z,Z),
=
r - 1
where
t
is an integer such that
f.
is that
(t+c)tf,
Zt + c
Z
C(k-1)(k-5)(k -4k-1) ,
Z
1C(k-1) (k-5) (k +3)/16
divides
k
t
1 (mod 4),
k- 1 (mod 4).
t
A consequence of Theorem 4 is the
COROLLARY.
Let
k >- Z
In order that
r,
since for any fixed
graphs
G(r,k).
is true iff
G(r,k)
r
be the class of all strongly regThen i f
k i: 5,
G(k)
for an infinite number of values of
there are a finite number of non-isomorphic
~
By Theorem 4, this can occur iff
vanishes, which
= 5.
k
We list below all the values of
r
which satisfy
~
r
necessary conditions of Theorems 3 and 4, for each value of
range
of
2
c, f
~
k
10.
~
The prime power factorization of
are also given in the table.
an asterisk.
is finite.
be infinite, it is necessary and sufficient
G(k)
that there exist a graph
G(k)
Z
Pn = 1.
I
PII = k-Z,
ular graphs with
PROOF.
and let
Note that for
a power of two and
c,
hence
k
=3
k
and the
k
and the values
~
Known graphs are marked with
there are no graphs, since
2t + c,
in the
is odd.
~
is
This result was obtained
by Erdos, Renyi, and Sos [5], and a different proof appears in Bose and
Shrikhande [4].
Note, also, that for the cases
k
= 4,
k
= 7 there is
18
at most one possible value of
r.
We may pose as unsolved problems the
determination of further necessary conditions for the existence of
G(r,k)
and/or the construction of any of these graphs, apart from the
three presently known, which are unique.
Values of
k
satisfying Theorems 3 and 4,
2
k
~
Possible values of
~
10.
r
c
f
2
1
1
3 • 5
3
1
2
2
4
1
3
3
7
5
2
1
0
s
6
1
5
5 • 11
7
1
6
24 • 3 • 5
337
8
1
7
3 • 7 • 31
85, 1681, 82405
9
2
2
24 . 3 • 7
31, 97, 127, 391, 1567, 56447
3
1
32 • 5 • 59
55, 505, 7831, 21755, 195805, 1762255.
10
I
r, k
·1
~
------
6.
2*, 3*, 7*, 57
4
none
I
2
,
where
s
-
-1, 0, 1 (mod 5)
I
31, 151, 3781
MOORE GEOMETRIES.
We may give a geometric interpretation to
2
Theorem 2, characterizing strongly regular graphs with
P11
=
1.
Let there be two types of elements 'points' and 'lines', and a
relation of 'incidence', such that a point mayor may not be incident
with a line.
integers
Then we have a Moore geometry
r, k
such that
r
~
k
~
2
MG(r, k, 2)
if there exist
and the following axioms hold.
(1)
Each line is incident with
(2)
Each point is incident with
k
r
points,
k
~
2.
lines,
r
~
k.
19
(3)
Any two distinct points are incident with at most one line.
(4)
Given two non-collinear points
line incident with
with
y
x,
x
and
y,
there is a unique
which intersects a line incident
(two lines may be said to intersect if they are
incident with a common point).
Then we get a graph
G(r,k)
from a Moore geometry
MG(r, k, 2)
by identifying points with vertices, and edges with pairs of points incident with the same line.
The set of points incident with a fixed line
clearly constitute a 'maximal clique'.
The axioms (1),(2),(3),(4) corre-
spond to the conditions (Ql),(Q2),(Q3),(Q4) of Theorem 2.
Properties (P3) and (P4) of Section 3, then show that there exist
no triangles and quadrilaterals in
The geometry dual to
MG(r, k, 2).
MG(r,k,2)
defines a graph
G*(k,r)
whose
vertices are the points of the dual geometry with collinearity defining
the adjacency relation.
ular except when
three.
The 'dual graph'
r=k=2.
G*(k,r)
is not strongly reg-
It is, however, metrically regular of diameter
The parameters of the corresponding three-class association scheme
can be determined directly in terms of
rand
k.
While this fact might
conceivably yield new necessary conditions for the existence of
G(r,k),
in terms of parametric identities and eigenvalue multiplicities valid for
three-class schemes, our calculations (not explicitly reproduced here)
show that no new necessary conditions are obtained beyond those derived
in Section 5 from analogous arguments for the original two-class scheme.
In general, one may define a 'weak geometry' of type
axioms (1), (2), (3).
r, k
by
A weak geometry may be said to be of diameter
if a sequence of at most
d
lines is required to join any two points
subject to consecutive lines of the sequence intersecting.
The number
d
20
of points in a weak geometry of diameter
1 + r(k-1) + r(r-1) (k-1)
(6.1)
We can define a Moore geometry
weak geometry of diameter
upper bound (6.1).
are points of
points of
d
MG(r, k, d).
are the graphs of
+ ... + r(r-1) d-1 (k-1) d .
MG(r, k, d)
of diameter
MG(r, k, d)
d
as a
is a graph whose vertices
and whose edges are pairs of collinear
Then the Moore graphs of diameter
graphs of Moore geometries
G(r,k)
is then bounded above by
for which the number of points attains the
The graph of
MG(r, k, d)
2
d
M(r, 2, d),
MG(r, k, 2).
d
are the
and the strongly regular graphs
Moore geometries of diameter
d
and the associated graphs will be studied in a subsequent communication.
21
REFERENCES
[1]
Bose, R.C.:
Strongly regular graphs, partial geometries, and
partially balanced designs;
Pacific J. Math., 13 (1963),
389-418.
[2]
Bose, R.C. and D.M. Mesner:
On linear associative algebras
corresponding to association schemes of partially balanced
designs;
[3]
Ann. Math. Statist., 30 (1959),
Bose, R.C. and T. Shimamoto:
21-38.
Classification and analysis of
partially balanced incomplete block designs, with two
associate classes;
J. Amer. Stat. Assn.,
~
(1952),
151-184.
[4]
Bose, R.C. and S.S. Shrikhande:
Graphs in which each pair of ver-
tices is adjacent to the same number
d
of other vertices,
Institute of Statistics Mimeo Series No. 600.6,
1969.
[5]
April,
(Submitted for publication.)
Erdos, P., A. Renyi, and V.T. S6s:
On a problem of graph
Studia Scientiarum Mathematicarum Hungarica,
theory,
1 (1966),
215-235.
[6J
Hoffman, A.J. and R.R. Singleton:
2 and 3;
On Moore graphs with diameters
IBM J. Res. and Develop.,
i
(1960),497-504.