0 7 8 9
6 1 7 8
5 0 2 7
4 6 1 3
0
7
8
9
1
3
5
2
4
6
6 5 4 9
1 0 6 5
7 2 1 0
8 7 3 2
9 8 7 4
2 9 8 7
4 3 9 8
3 4 5 6
5 6 0 1
0 1 2 3
8 7 1
9 8 2
6 9 3
1 0 4
3 2 5
5 4 6
7 6 0
0 1 7
2 3 8
4 5 9
1 3 5 2
9 2 4 3
8 9 3 4
7 8 9 5
8 6
2 3
7 0
3 4
6 1
4 5
0 7
5 6
1 9
6 0
2 8
0 1
1 2
4 6
5 0
6 1
0 2
1 3
2 4
3 5
8 9
7 8
9 7
COM BIN ATOR 1 A L
MAT H EM A TIC S
YEA R
8 9
9 7
7 8
Febrruary 1969 -June 1970
SOt1E FURTHER CONSTRUCTIONS FOR
G2 (d)
GRAPHS
by
R.C. Bose "
University of North Carolina
5.S. Shrikhande
University of Bombay
Institute of Statistics Mimeo Series No. 600.16
Department of Statistics
University of North Carolina at ChaDel Hill
I
,,-
;l)vEr~BER
1969
'This research Was supported b)./ the U. S. Air Foree OfFice oj' Scient-i fie Research
un(~r Grant No. AFOSR-68-1406 and the National Science Foundation under Grant
No. G2-8624
SOME FURTHER CONSTRUCTIONS FOR G (d) GRJUlHS
2
by
R.C. BOSE, UNIV8RSITY OF NORTH CAROLINA,
C,IAl'ZL HILL, N.C., U. S .A.
&
S .S. S"rlRI KHANDE , UNIVER3ITY OF BOMBAY,
BOMBAY, INDIA.
ABSTRACT
A G2 (d) graph is a finite, undirected graph without loops or
mul tiple edges in which each pair of ve rtices is ad,j acent to exactly
other vertices, d
in
L-2-!.
~
2.
d
An infinite family of such graphs was given
The present paper gives some further constructions for these
graphs.
1. KNOWN RESULTS
'fie use the notation and terminology of
L-2-!
and quote some
results contained in this paper.
THEOREM A.
that v _ 1
f\
=n 1 (n1
G (d) graph, d ~ 2,is regular of valence n such
1
2
- '1 )/d where v is the number of verticesrd thore
exists a positi ve integer m such that
n
(ii)
1
=d
2
+ m ,
and
dim is an integer with the same parity as
v - 1 - m.
note that a G (d) graph with parameters (v, n, d), d ~ 2,
2
2
is essentially a strongly regular graph with parameters (v, 01' P1\' P1 1)
fie
where P1\
= P~1 = d.
By a pseudo L/k) graph we will mean a pseudo net
graph L (k) and by an 1TJJ (k) graph ...a negative Latin Square NL (k) graph,.
r
r
~
r
2 THEOREM B.
The existence of a pseudo L
r
1
(2r ) and a pseudo L (2r )
1
r
2
2
graph implies the existence of a pseudo L ( 2r) graph wi. th
r
THEOREM C.
The existence of a pseudo L
r
1
r::z 2r r •
1 2
(2r ) and a Nt (2r )
1
r
2
2
graph implies the existence of a NL (2r) graph with r = 2r r •
r
1 2
I
TI{E()ffi,~ D.
~se\ldo L (2r) ~d Nt (2r) graphs exist' for
r
r
m m+n_1
)
r ;: :3 .2
, where m, n are nonnegative integers (m, n
I
(0, 0).
Noting that a pseudo L (k) graph is a strongly regular graph
r
2
with parameters (k , r(k-1), k -
2+
(r - 1)(r -
2),
r(r - 1»
and a
2
NL/k) graph is strongly regular with parameters (k , r(k + 1),
-k - 2 + (r + 1)(r + 2), r(r + 1»
THEOREM E.
we have
A G/d) graph with parameters
2
(i) v= 4r , n
(11) v= 4r2 ,
(iii) v= 4r
2
1
;: r( 2r - 1), d = rCr - 1),
n
1
= r(2r
2
- 1, n 1 = 2r ,
+ 1), d
d
= r(r
+ 1 ),
= r 2,
...m m+n-1
exists for aJ.l r = :> .2
, where m, n are non negative integers
(m, n)
J (0, 0).
2. NEW CCNSTRUCTIONS
We first prove some preliminary results.
Larmna 2.1.
vr = bk.
Let v, b, k, r be non negative integers, k
~
v and
Let there exist an incomplete block design D with v symbols
(treatments) in b subsets (blocks) of size k such that any pair of
3 treatments occurs together in at most one block.
Then a necessary Eftd
sufficient condition that each block in D intersects precisely k(r - 1)
other blocks in D is that each treatment occurs exactly r times "in D.
Proof.
It is obvious that arr:r two blocks in D intersect in at most
one treatment.
If each treatIoont occurs r times in D, then any block
containing, say, treatments t , t , ••• , t intersects (r - 1) other
1
2
k
blocks containing t .. ; i
~
taining t
i
and t
= 1,
2, , •• , k.
The sets of (r - 1) blocks con-
are obviously disjoint, i ~ j.
j
Hence this block
intersects precisely k(r - 1) other blocks necessarily in one treatment.
This proves the sufficiency part of the theorem.
Now suppose that each block intersects precisely k(r - 1)
other blocks.
D.
Let r. be the number of times the treatment i occurs in
~
Then obviously
r
=
",
~ri/V
= r.
Let N = (n1j ) be the usual. (0, 1) incidence matrix of D with v rows
and b columns where nij = 1 or
~\O(.~
~s
j or not.
° according as treatment i
occurs in
Then from our hypothesis
,
N N = kl b + A
where A is an adjacency matrix of order b which is regular and of
valence k(r - 1).
Also
where B is also an adj acency matrix of order v with i-th row sum
.-
4
=
tr ((NN' )(NN'))
=
But
=
tr (N(N'N)N')
=
tr N(kl
= k
tr NN' + tr NAN'
=k
tr NN' + tr A N'N
=k
tr NN' + tr (A(kl
=k
~ri
,
tr (NN' )(NN'))
=k
-
v r
b
+ A)N'
b
+
A»
2
+ tr A
+
bk(r-1)
= v r(k + r - 1).
Hence equating the two values of the trace
_ r)2 =
=
=
which implies that each treatment occurs r times in D.
This canpletes
the proof of the 1 emma.
We now define the concept of an ascendent graph G* of a
strongly regular graph G with parameters (v, n ,
1
(V , V2) be a partiti:tm of the vertex set V of G
1
respectively ccntain
nt and v - nt vertices.
Let en be a vertex
not in V and let G* be a graph with vertex set ( <X:
adjacency in G* as follows:
of V
1
0
U V) °
Ve define
The vertex CO is adjacent only to vertices
If x, yare in V, then they are adjacent in G* if and only if
-
5
they are adjacent in G and belong bci;h to V or both to V , or it
1
2
they are
nonadj acent in G and belong one to V ::nd the other to V2.
1
* n * , P11'
1* P11
2*)
If the graph G* is strongly regular with parameters ( v,
1
where Y* is necessarily v + 1, then G* is said to be an ascendent of G.
We derive the conditions under which a graph G with parameters
(v, n 1 ,
P~1'
P;1) has an ascendent G* with parameters (v*, nf'
p~~, p~~).
We will assume that G is neither a void graph nor a complete graph i.e.
If G* is an ascendent of G, then G is a descendent of G* with
respect to the vertex 00 and hence from
L2J
- 6
-
Hence from the usual parametric relation
which implies that
nz!2
1
= P12
1
= P22
Also from (2.1), (2.3) and (2.4)
,
2
v + 1 = 4n, - 2p'1 - 2p1'
or
v=
,
2
6P11
2p'1
-
1•
Thus (2.7) is a necessary condition forG to have an ascendent.
1
2
It is easy to see that for a graph G with parameters (v,n 1 , P", P11)
each of (2.5), (2.6), (2.7) implies the other two.
L-2..!
It also follows from
that the vertex set V of G can be partitioned into (Vl' V2) with
n~ vertices in V, and v - n~ vertices in V2~ where the set V1 is the set
of vertices in G* which are adjacent to 00 •
h
*
,
is adjacent to P11 = n, - n, + P"
V
2
.
Further, each vertex in V1
vertlces of V, in G and each vertex in
is adjacent to
vertices of V in G.
2
CpnverselY"suppose (2.7) is satisfied for G and further there
exists a partiilion
(v"
V ) of the vertex set V of G with n~ vertices in
2
V, and v - n; vertices in V ' such that each vertex in V, is adjacent to
2
-
7
1
. adjacent to P11
2
n * - n 1 + P11vertices
in V1 and each vertex in V2 ~s
1
vertices in V •
2
Then by using azguments similar to those in £"2.../,
CAr
it can be shown that G* isLstrongly regular graph with parameters (2.2),
(2.3), (2.4) provided
p~~, p~7
are non negative integers.
From the
relation
it follows that n* satisfies the equation
1
The above equation is easily seen to have real positive roots.
Since n~ is necessarily an integer a further necessary condition for G
to have an ascendent G* is t hat the above eEluation has an int egral solution.
We easily verity that
o
f (n1 - P;1)
= P~1 (p~, -
Further, it is easily seen that if
1) •
P~ 1 = 0,
then f(x)
=0
has no integral
Hence, if the equation has en integral solution then we can
solution.
assume that
P~ 1 ~ 1 and then the above relations imply that
2*
*
2
P.,., = n 1 - n 1 + P 11
)
o.
-
8
Thus the c:mdition that rex) = 0 has an integral oolution n~ is neces-
sary and sufficient for
1*
2*
nOnnegativeness of P11' P11'
We can, therefore,
state the following theorem.
THEOREM 2.1.
(v, n 1 ,
Let G be a strongly regul ar graph with parameters
P~1' P;1)'
Then G has an ascendent G* with parameters
1* P 2*) if and only if the following parametric and structural
( v* • n * , P11'
1
11
conditian~(P) and (S) are satisfied in G.
(p)
2
1
v = 6 P11 - 2 P11 - 1.
(8)
The equation
2
1
2 )
x + x (P 11 - 5 P11
2
+ v P11
= o.
has an integral solution n~ and there exists a partition (V , V2) of
1
the vertex set V of G wi th n~
vertices in V1 and v - n~
vertices in
1
V2 such that every vertex in V has n *
1 - n 1 + P11 adjacent vertices in
1
2
V and every vertex in V has P1 1 adjacent vertices in V •
1
2
2
The parameters of G* are then given by
v * ;: v + 1,
n *,
1
o.
It is obvious that any two blocks of a BIBD wi th ~
have at most ane treatment in common.
A = 1.
b ;: 4k
2
Consider a BIBD with r
Then the values of v and b are given by v
- 1.
= 2k 2
=
=
2k + 1,
- k and
Consider the blocks as vertices of a graph G and define
two blocks as adjacent or nonadjacent. according as they have a treatment in commn or not.
Then
L2J G is
strongly regular with
-
9
parameters ( 4k 2 - 1, 2k 2 , k 2 , k 2 ) and satisfies the condition (p )
of the above theorem.
Also the eqUati;n f(x)
tions k( 2k - 1) and k( 2k + 1).
=0
has .integral solu-
Take n7 ;;:: k( 2k - 1).
If the 4k
2
- 1
blocks can be partitioned into sets V and V2 of k( 2k - 1) and
1
(k + 1)(2k - 1) blocks respectively such that each block in V is
1
adjacent to k
k
2
2
- k blocks in V and each block in V is adjacent to
1
2
blocks in V , th8n the condition (S) is also satisfied.
2
From
Lemma 2.1 this means that the set V (respectively V ) contains each
1
2
of the 2k
2
- k treatments exactly k (respectively k + 1) times.
We note that a BIBD with r = 2k + 1,
geometry (r, k, t)
=
(2k + 1, k, k).
A
= 1 is a partial
The graph G. 1s then
L1J the
graph of the dual configuration and is also a partial geometry
(k, 2k + 1, k).
THEOlTh"'M 2.2.
r
=
2k + 1,
We can, therefore, state the following theorem.
Let G be the graph of the dual of a BIBD with
)..::: 1.
Then G has an ascendent G* which is a pseudo
~.( 2k) graph if and only if the 4k 2 - 1 blocks of the BIBD can be
partitioned into sets V and V of k(2k - 1) and (k + 1)(2k - 1)
1
2
2
blocks respectively such that each of the 2k - k treatments of
the BIBD occur k times in V and k + 1 times in V •
2
1
BIBD's having the structure of the above theorem exist for
k ::: 5 and 7.
See for example Appendix I in
L-4J.
Hence we have the
following result.
COROLLARY.
Pseudo L (1O) and pseudo L/14) graphs exist.
5
Goethals and Seidel
L-3...!
graph in precisely the same manner.
have constructed a pseUdo ts(1O)
1']
Using the Corollary and Theorems B, C and D we have the
following theorem
THEOREM 2. 3 •
m 5a
r -_ 3
7c
(I)
pseVdo L (2r) graphs exist for all
r
2m+a+c+n-1
wh ere m, n, a, c are non negat i ve integers
(m, n, a, c) f. (0, 0, 0, 0).
(II) NL r (2r) graphs exist for r = 5
a
m 5a
nega:ti ve i n t egers an d f or r -_ 3
are non negative integers and em, n)
7
c
7c
a c
2 +
h
were,
a, c are non
2m+a,tlc+n-1
f. (0,0).
The proof is similar to that of Theorem
L-2-!
where m, n, a, c
9~3
and 9.5 in
and is omitted.
Finally, noting that pseudo L (2r) and NL (2r) graphs are
r
r
G ( d) graphs we have
2
THEOREM 2.4.
. +
G (d) graphs with the following parameters exJ.s"
2
2
(i) v= 4r ,
n
1
(ii) v= 4r 2 - 1,
= r(2r - 1),
n
1
2m+a+c+n-1
1-== m Sa
c
for all if> 3
7
=
2
2r ,
d=r(r-1);
d
= r 2;
where m, n, a, c are non negative
integers (m, n, a, c) ~ (0, 0, 0, 0).
n
with r
=
2
m 5a
wi th r = 3
7°
a+c
1
= r( 2r
+ 1),
d
= r(r
+ 1);
, where a, c are non negative integers and
2m+a+c+n-1 where m, n, a, c are non negative
integers and (m, n) ~ (0, 0).
We remark
that since our construction is essentially by a
composition method, any new G (d) graph can be utilised in cmjul1ction
2
with the above theorem to enlarge B"ch a family considerably.
-\\ REFERENCES
1..
BOSE, R.C.,
"Strongly Regular Graphs, . partial geometries 3lld
partially balanced designs," Pacific J. Math. 13(1963),
pp. 389 - 418.
~.
BOSE, R.C. AND SHRIKHANDE, S. S., "Graphs in which each pair of
vertices is adjacent to the same number d of other vertices, "
To appear in Studie Scientiarum Mathematicarum Hungarica.
3.
GOETHALS, J .M. AND
SEIDEL, J.J., "Strongly regular graphs derived
from combinatorial designs,
If.
II
To appear in Canad. J. Math.
Hall Marshall, "Combinatorial Theo:ry," Blaisdell( 1967).