Introduction. Given a set V= {x,, x2, . . . , G} of v points we have a two-class association scheme if there is a relation of association between any two distinct points.
Col~mda StatekMver&y, EW Coliins, C~iorado, U.S.A. M.Sa SHRIKHANIXJ Pddaoi VflJosrslty, Shim, Iran, Received 4 April 1978; revised manuscript received 16 May 1978 Recommended by A, 3. Hoffman AMtact: -Bos&and Clatwottihy (1955) showed that Me parameters of a two-class balanced incomplete block design wit&Jk,= 1, A, = 0 and satisfying I 9 &can be expressed&interms of just three pariameters r, k, k Later Bose (1963) shoJved that such a design is a partinl geometry (r, k, t). Bose, Shrikhande and\ Singhi (1976) have defined partial geometric ..LIgns ir, k, t, c), which reduce to partial geometries nhen c = 0. In this note we prove that auy two class partially balanced @BIB) design with r < k, is a partial geometric design for suitably chosen t, k, t, c and express the parameters of the PBIB design in terms of r, k, 1,c and A,. We also show that such PBIB designs belong to the class of special partially bclanced designs (SPI3IB) studied b) Bridges and Shrikhpnde (1974).
1.Introduction Given a set V= {x,, x2, . . . , G} of v points we have a two-class association scheme if there is a relation of association between any two distinct points satisfying “the following conditions: (a) .Any twa poitlts are either first associates or second associates. (b) Each point has ni ith associates (i - 1,2). (c) If two points are ith asiiociates, then the number of treatments common to the jth associates of the firgbtand kth associates of the second is pfk and is independent of the pair of points with which we start, Also pjk= &. If we assume the constankcyof II, nl, p il and p& then the constancy of p&, piI, p&, p$, p$, p& follows and pi2 = p&, p& = & [Bose clnd Clatworthy (19533. The concept of an m-class zissocbationscheme was first introduced by Bose and Shixnamoto (1952). WC:need only the special case m = 2 for the purposes of this
paper. a’The research of this author was supported by AFOSR grant no. 77-3127, 91
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A graph P (without loops or mu1tiple edges) with P) vertices is said to be strongly regular (Bose, (1963)] if (i) P is regular of degree nl. (ii) Any two vertices of P are shnultaneous1y adjacent to pi1 or p& other vertices of P according as they are adjacent or non-acIjgcent* We may identify the o points of a two-&&’ associtition’ scheme”with’,the o vertices of a graph P, and consider two vertices adjacent or non-adjacent according as they are first associates or second associates. It is clear that a strongly regular P with parameters
(1.1) is isomorphic with a two-class association scheme with the same parameters. The adjacency matrix of the graph P will also be denoted by P. If we have a two-class association scheme or equivalently a strongly regular graph P with parameters (l.l), then we get a partially balanced incomplete block (PBIB) design D based on P, if we can arrange the o vertices (points) into b subsets (blocks) such that (a) Each bloclc contains k points (all different), (@i Each point is contained in r blocks, (~11if any two points are ith associates (i = 1,2) then they occur together in Ai blocks. Then
are said to be the parameters of D. Note that v occurs both in (1.1) and (1.2). There are a number of well-known relations among these parameters [Bose and Nair ( 1939)]. PBIB designs were first studied by Bose and Nair (1939), though the concept of association schemes was not explicitly introduced in this paper. For further background on partialklybala.nced designs and association schemes see Nair and Rao (1942); Bose and Comror (1952); Bose, Shrikhande and Bhattacharya (!953), Bose and Clatworthy (1955), Bose and Mesner jlPS9j and Dembowski (19G3, pp. 281-291, 294, 303, 317). Bose (1963) defined a partial geometry (r, k, t) as follows. We have a system of undefined points and lines together with an inci&nce relation satisfying the following axioms: Al: tiny two points are incident with not more than one line. A2: Each point is incident with t lines. A3: Each line is incident with k points. A4: If the point P is not incident with the line I, there pass through P exactly t lines (2a 1) intersecting I, By taking the points of the partial geometry as the points and the lines as the
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t -=t -k[
(r-l)(k-l)+t],
B=;*
rJC,Al=l,
A&l
a31
"
an,“_the *3 strongiy regular graph~”(associtition -based I ,” 1 ’ * scheme) P with parameterst j ,’
u, np
r(k4),
*
&1=(t-l)(t-l)+k-2,
$1 = rl (letcr,
lgtsk).
(1.4)
gusJ a pa~tii~ geometry (r, k, t) is isomorphic to a PBIB design (u, b, 5 kj A;, Az= 0) based on’the graph .,of the geometry. However, an arbitrary PBIB design .based on a strongly reiular graph is not necessarily a partial geometry. Bpse a$ Clatworthy (1955) showed that if there exists a PBIB design - /-/I El(ti, >b,r, k, Al = i, Aa= 0) based on a strongly regular graph P(t), q, pi 1, pf ,) for which t < k, then parameters of 13 and P are given respectively b:y(1.3) and (1.4). Irt other words, the PBIB design has the same parameters as that arising from a partial..geometry (1; k, t). Bose (1963) proved later that the PBIB design is, in fact, a partial geometry (r, k, t). In this note we, shall extend the result given above to arbitrary two-class PBIB designs, Al> AZand r 1
