Tilburg University

Strongly regular graphs induced by polarities of symmetric designs

Haemers, W.H.; Higman, D.G.; Hobart, S.A.

Publication date:1990

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Citation for published version (APA):Haemers, W. H., Higman, D. G., & Hobart, S. A. (1990). Strongly regular graphs induced by polarities ofsymmetric designs. (Research memorandum / Tilburg University, Department of Economics; Vol. FEW 452).Unknown Publisher.

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STRONGLY REGULAR GRAPHS INDUCED BYPOLARITIES OF SYNIly1ETRIC DESIGNS

W.H. HaemersD.G. Higman, S.A. Hobart

FEW 452

STRONGLY REGULAR GRAPHS INDUCED BY POLARITIES OF SYMMETRIC DESIGNS

by

W.H. HAEMERS,Tilburg University, Tilburg, The Netherlands,

D.G. HIGMAN,University of Michigan, Ann Arbor, Michigan, USA,

and

S.A. HOBART,University of Wyoming, Laramie, Wyoming, USA.

ABSTRACT. We study symmetric designs with a polarity for which the graphinduced by the absolute points and the graph induced by the non-absolutepoints both are strongly regular. The main result ís that all parameterscan be expressed in terms of only one parameter.

z

INTRODUCTION.

Throughout, we shall assume that A is the (0,1) incidence matrix of asymmetric 2-(v,k,a) design D with a polarity with v(~ 0) absolute pointsiand v- v-v (~ 0) nonabsolute points. So A takes the form:z i

A -r A C 1

Ill~ JCT A

2

where A1 and Az are symmetric matrices, A has ones on the diagonal and AZrhas zeros on the diagonal. Thus A-I and A are adjacency matrices of~ 1graphs I' and i' , say. We call i and i" the graphs induced by the polarityof D. 1'he above decomposition of A is calle~d the polar deeomposttton . Apolar decomposition is regular if I' and I'Z are regular, and strongly

reguZar if I'1 and I"2 are strongly regular, complete or empty. The incidencestructure of absolute points and nonabsolute blocks, given by the matrix C,is called the polar structure of the design D.We assume that D is nontrivial, i.e. 0~ a C k-1 ( v-2. PolardecomposiY,ions of trivi-al designs are obvious and not very interesting.

In the present paper we study strongly regular polar decompostions.It is, in a certain sense, an extension to a paper on strongly regulargraphs wit.h strongly regular decompositions by t.he First two authors [1].1'he sub,jecl. iv motivrif.ed by t.he t'ol lowinq examplc~. I,eL D be Lhe design oft,o f n i v nnrl I,yl,c~i,~, I nnc~v ol' I'(1( r~ ,~I )( u ) 2). 'I'hc~n Lho grnt,l,4 l nducc:d by Lhc~unitary polarity are strongly regular (see for instance [3]). Theparameters for this example are:

~- 4n}1-1 . k- 4n-1 I~- 4n-1-1 , V - 22n.1-(-2)n-13 3 3 ~ 3 '

We shall prove that all parameters of a strongly regular polardecomposition can be expressed in terms of only one parameter a. If a is apower of -2 we find the values above, but unfortunately, other values of afor which no examples are known, remain feasible.

3

For strongly regular I' (i - 1,2) the parameters are denoted by v, k', a1 ~ ~ ;and ui, and the eigenvalues of Ai by k, p and oi, such that ~Pí~ ) ~a ~.Note that k- k'tl and k- k:. The multiplicity of a is denoted by pi.

If fis complcte or c~mpty we put: g~ - v-1 (p disappeares).~ I I I

PRELIMINARIES.

The following three Lemmas are the analogues of 2.2, 2.4 and 2.5 of [1].Since the proofs are the same, we omit them here.

Lemma 1. Suppose I'1 is regular of degree ki - kl-1. Then

k v-v k1 1v-v i

C k~ .

Equality holds if and only if the polar decomposition is regular.

Note that for the polar decomposition to be regular k-a (the order of D)must be a square.

For a regular polar decomposition we define k2 to be the degree of I'z and

(1)k v-v ki ia - .v-vi

Then we easily have

(2) a- k~ tkz-k - tk~, ~al ~ 2.

So a, -a and k are the eigenvalues of A with multiplicities ~p (say), v-1-pand 1, respectively.

Lemma 2. Suppose A~ has eigenvalues kl, 61 and pl (loll C ~pl~) with

multiplicities 1, ~ol and vl -1-g~1 (1 (~1 ( v~ -1) , respectively. Let the

q

polar decomposition be regular. '1'hen A has eigenvalues k,-o ,-p , a, -a1 1 1 1

with multiplicities 1, pl, vr-l-pl, p-vr and vz-p, respectively.Also ~6 I~ ~Pr ~~ ~a~ . and ~o ~~ ~a~ .

Lemma 3. With the hypotheses of Lemma 2, the polar decomposition isstrongly regular if and only if one of the following occurs.i. v - v - p,

r zii. p- a, v- p,i iiii. pi - -a, vz - p,

It Ia en5,y to 5er: l.hnt tf f1 occur4 for m m~trix A, Lhen 1ti occurs for thccomplementary case, that is for the design with incidence matrix ( J denotesthe all-one matrix).

J -l AZ CTJ ~C A1

Note that Lemma 3 also applies if I'1 is complete or empty (the conditionsp- ta are meaningless ín this case). Next we show that case i does noti -occur.

Lemma 4. A strongly regular polar decomposition with vl - vz - p does notexist.

Proof. Assume v~ 2k (otherwise consider the complement). Trace A- vl -~p- k t~pa - ( v-p-1)a implies k- vr-a. Hence, by (1) and (2), kl - kz - v1~2- vz~2. Therefore, since i'z is strongly regular, kz(kz-~z-1) - uz(vz-kz-1)- u(k -1). This implies N. - 0 mod k, hence I" is complete bipartite, soz z ~ z zp, --k . This i s ímpossible , since then, by Lemma 2, ~a~ ~ ~P I- k -z z - 2 zvz~2 - v~4, which is a contradiction to aZ ( k C v~2 and ~a~ ~ 2. Q

5

MAIN TREOREM.

Theorem 5. If D has a strongly regular polar decomposition thena- 4 mod 6, and, up to taking complements, all other parameters areexpressible in terms of a as follows:

v - 3(16a~-1) , k - 3(4az-1) , ~ - 3(az-1) ,

Pi

~~ - ~ - 3(2ati)(4g-i) , ~Z - 3a(4a-1) .

k~ - k-1 - 3(a-1)(af2) , k1 - kz - 3a(2a-1) ,

-pz - a c~ - -o - az i 2 '

P~ - f~2 - 9(a-1)(2a}1) .

ul - b(a-1)(at2) , at - b(a2t4a-2) .

~z - ba(at2) , a2 - ba(a-1).

Proof. We may restrict ourselves to case ii of Lemma 3. By this result wecan express all parameters in terms of k and a only. We have a- k-az, andhence, using ~(v-1) - k(k-1):

v - k` -a~

k-a`

Trace A- v - p- k.~oa -(v-1-~p)a yieldsi

k(kta)(a-1) a(kta)(ktl-2a)(k-az)(2a-1) ~ z (k-a2)(2a-1)

Next, using (1) and (2), we find

ak k(atl)(a-1)`kz - 2a-1 ~ vZ-kz-1 -

(k-a~)(2a-1).

6

Since k-a~ - a) 0 and ~a~ ) 2 it follows that 0 C k C v-1, so I" is not- ~ z zthe empty graph or the complete graph. Moreover, 2a-1 divides k. Definex- k~(2a-1), then k- ax and a and x have the same sign. By use ofzpz - -a (by Lemma 2 and 3) and (k -p )(k -o ) - (k tp d )v (because I' isz z z z z z z z zstrongly regular) we find after some computation:

6

Define

a(axta-2x)z - 2ax-x-1 '

d - 4a -2af3 - ~a2tza-3x-3z 2ax-x-1 '

then d) 0, since d C-1 would imply (2e-1)(afl) C x(1-a) C 0, whichcontradics ~a~ ) 2. Moreover, using 2ax-x-1 - k-1-) az, we have

d C~} a 2- xtl C 4 ~ á C 5 if a and x are positive,- a 2ax-x-1 - ~-

4 2a-1 C 5 if a and x are negative.

Since d is odd we can conclude that d- 1 or d- 3. If d- 1, then x- 2a-1and ~1 -(za-i)(4a`-3at1)~(3a-1), which is no integer. Hence d- 3, so

x - 2af 1 a3 ' 6z - 2 , a-4mod6.

All other parameter now follow in a straightforward manner. 0

If a-(-2)n-1 we find the parameters of the example given in theintroduction. No other examples are known. This leaves a- 10 (D is a2-(533,133.33) design) as the smallest unsolved case. By Theorem 5, onlyfor a--2 one of the graphs induced by the polarity is complete or empty.For a--2, however, the structure is unique. So YG(2,~1) with its unitarypolarity and the complement provide the only strongly regular polardecomposition for which one of the induced graphs is empty or complete. Inthis case the polar structure is AG(2,3), the affine plane of order 3. If~a~ ) 2, it can be shown (by simular, but less complicated, arguments asfor Theorem 2.8 of [1]) that the polar structure is a strongly regulardesign (see [2] for definition and notation) with parameters

7

(3) - (v ,k-k ,a-2-X ,a-N .v ,k-k a-a ,a-u.,),i i i i ~ z' ~ ~

and t}iat, conversely, a strongly regular design whose parameters satisfy(3) with v, k, k etc. as given in Theorem 5, is the polar structure ofsome symmetric design with a strongly regular polar decomposition.

A symmetric design with a polarity can be seen as a strongly regular graphfor which loops are admitted. More precisely, i f we allow loops in graphs,a strongly regular graph is a simple strongly regular graph or a symmetricdesígn with a polarity. In this more general setting, a strongly regulargraph with strongly regular decomposition either belongs to the casewithout loops treated in [1], or is a design with a strongly regular polardecomposition. This can be proved as follows. For a strongly regulardecomposition of a symmetric design with a polarity it follows byeigenvalue arguments as in Lemma 3 , that if I'i is a symmetric design with apolarity, then so is I' . This, however is impossible by Rahilly [4]. Thuszboth I' and I' must be strongly regular graphs without loops, so we have ai 2strongly regular polar decomposition.

REFERENCES

[1] W.H. Haemers and D.G. Higman, Strongly regular graphs with stronglyregular decomposition, Linear Algebra App1. 114~115: 379-398 (1989).

[2] D.G. Hígman, Strongly regular designs and coherent configurations oftype (S .i) , European J. Combin. 9: 411-422 (1988).

[3] X.L. Hubaut, Strongly regular graphs, Discrete Math. 13: 357-381(1975).

[4] A. Rahilly, Divisions of symmeY.ric designs i nto two pnrts, Graphs andComb i nntor i cs h: fi7-73 ( l~)FiFi ).

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