Symmetric Balanced Incomplete Block Design Research Articles

Construction of balanced incomplete block designs

AbstractGiven a symmetric balanced incomplete block design or a finite plane, we recursively construct balanced incomplete block designs by taking unions of certain blocks and points of the original design to be the blocks of the new design.

Symmetric group divisible designs with the dual property

A symmetric group divisible design (SGDD) is said to have the dual property, if the dual of the design has the same parameters as the original design. In this paper we study the structure of such designs and give some applications to Baer subdesigns of symmetric balanced incomplete block designs (SBIBD) and to near planes (Fast-blockpläne).

On (v, k, λ)-configurations with v=4pe

A (v, k, λ)-configuration, also called a symmetric balanced incomplete block design, is an arrangement of v distinct objects called points or varieties into v subsets called lines or blocks such that each line contains exactly k points and each pair of distinct lines contains exactly λ points in common. To avoid certain trivial configurations, one assumes that 0<λ<k<v–1.

Embedding of quasi-residual designs

A simple sufficient condition for embedding of a quasiresidual balanced incomplete block design into the corresponding symmetric balanced incomplete block design is obtained.

On symmetric balanced incomplete block designs with doubly transitive automorphism groups

On symmetric balanced incomplete block designs with doubly transitive automorphism groups

Some notes on integral equivalence of combinatorial matrices

Recent work on integral equivalence of Hadamard matrices and block designs is generalized in two directions. We first determine the two greatest invariants under integral equivalence of the incidence matrix of a symmetric balanced incomplete block design. This enables us to write down all the invariants in the case wherek −λ is square-fre. Some other results on the sequence of invariants are presented. Secondly we consider the existence of inequivalent Hadamard matrices under integral equivalence. We show that if there is a skew-Hadamard matrix of order 8m then there are two inequivalent Hadamard matrices of order 16m, that and there are precisely eleven inequivalent Hadamard matrices of order 32.

On characterizing certain graphs with four eigenvalues by their spectra

The eigenvalues of a graph are the eigenvalues of its adjacency matrix. In this paper regular connected graphs with four eigenvalues, the least of which is — 2, are examined. Many line graphs are of this type and it is shown that, with only a finite number of exceptions, all graphs of this type are line graphs of strongly regular graphs. symmetric balanced incomplete block designs, or complete bipartite graphs. Certain of these graphs are characterized by their spectra among all graphs with the same number of vertices. We show that if the number of vertices is large enough, the existence of a graph with the same spectrum and number of vertices as the line graph of a complete bipartite graph and which is not isomorphic to it is equivalent to the existence of a symmetric Hadamard matrix with constant diagonal. Finally, we give some necessary and sufficient conditions for the line graph of a complete bipartite graph to be characterized by its spectrum and number of vertices.

Automorphisms of symmetric balanced incomplete block designs

Automorphisms of symmetric balanced incomplete block designs

On the orbits of collineation groups

In this paper we consider some results on the orbits of groups of collineations, or, more generally, on the point and block classes of tactical decompositions, on symmetric balanced incomplete block designs (symmetric BIBD = (v, k, 2)system=finite 2-plane), and we consider generalizations to (not necessarily symmetric) BIBD and other combinatorial designs. The results are about the number of point and block classes (or orbits, i.e. sets of transitivity) and the numbers of elements in these classes. In Sections 2, 3 and 4 below we exhibit the key role of the rank of the incidence matrix of a design, while the remainder of the paper uses more specific properties of the incidence relations. Included in Section 2 is a simple new proof of the theorem of DEMBOWSKI [7] on the equality of the numbers of

Some Methods of Construction of Designs for Two-Way Elimination of Heterogeneity—1

Abstract Two methods of construction of designs for two-way elimination of heterogeneity are given. The first method is based on the derived and residual designs of symmetrical balanced incomplete block designs. The second method utilises the balanced incomplete block design with parameters v = 2k, b = 2(2k − 1), r = 2k − 1 and λ = k − 1. It is further shown that the designs so constructed are either balanced or partially balanced.

On Dependent Tests from a Non-Orthogonal Design

Abstract This paper illustrates, through a special case, an effect of non-independent numerators on F-test's in an analysis of variance. Symmetrical balanced incomplete block designs are considered with “blocks” a random effect and no interaction. The joint density of the adjusted sums of squares is determined under the null hypothesis of no treatment effect. Given the result of testing for the block effect, the conditional probability of a Type I error when testing the null hypothesis concerning the treatment effect is evaluated for a number of designs.

On Dependent Tests from a Non-Orthogonal Design

Abstract This paper illustrates, through a special case, an effect of non-independent numerators on F-test's in an analysis of variance. Symmetrical balanced incomplete block designs are considered with “blocks” a random effect and no interaction. The joint density of the adjusted sums of squares is determined under the null hypothesis of no treatment effect. Given the result of testing for the block effect, the conditional probability of a Type I error when testing the null hypothesis concerning the treatment effect is evaluated for a number of designs.

On the Line Graph of a Symmetric Balanced Incomplete Block Design

On the Line Graph of a Symmetric Balanced Incomplete Block Design

On the line graph of a symmetric balanced incomplete block design

1. Introduction. We shall study the relations between an infinite family of finite graphs and the eigenvalues of the corresponding adjacency matrices. All graphs we consider are undirected, finite, with at most one edge joining a pair of vertices, and with no edge joining a vertex to itself. Also, they are all connected and regular (every vertex has the same valence). If G is a graph, its adjacency matrix A = A (G) is given by 1 if i and j are adjacent vertices, ai O otherwise. The line graph L(G) (also called the interchange graph, and the adjoint graph) of a graph G is the graph whose vertices are the edges of G. With two vertices of L(G) adjacent if and only if the corresponding edges of G are adjacent. There have been several investigations in recent years of the extent to which a regular connected graph is characterized by the eigenvalues of its

Classification and Analysis of Linked Block Designs

In comparative experiments involving a fairly large number of varieties, when for lack of homogeneous experimental units, complete block designs are not available. Balanced Incomplete Block (BIB) designs prove very convenient in two respects. Firstly, the analysis is very simple and secondly, comparison between any two varieties has the same precision. On the other hand, a BIB design requires a large number of experimental units, because balancing is not possible unless the number of blocks is atleast as large as the number of varieties. To obviate this dif ficulty, different types of incomplete block designs have been introduced, of which the most notable is the Partially Balanced Incomplete Block (PBIB) design introduced by Bose and Nair (1939). Another class of incomplete block designs, called Linked Block (LB) designs was introduced by Youden (1951) which he obtained by dualising several BIB designs, that is by taking the varieties and blocks of the BIB design respectively as blocks and varieties in the LB design. The LB designs so constructed by Youden happen to be all PBIB designs, but this is not necessarily true. A great advantage of LB designs is that the analysis can be easily worked out. In this paper, we show how the intra-blockTanalysis of a LB design can be neatly carried out. The efficiency factor of a LB design is obtained. The methods are illustrated with a numerical example. An exhaustive list of all LB designs with ten or less plots per block and involving ten or less replications is given. These designs fall into three groups: (1) symmetrical BIB designs, (2) PBIB designs and (3) Irregular designs For plans of LB designs belonging to group (2) reference is made to the serial number of the two-associate class PBIB designs enumerated by Bose, Clatworthy and Shrikhande (1954). Plans for other designs belonging to groups (2) and (3) are given in detail. Certain theorems are derived which are useful in determining from the parameters of a given two-associate class PBIB design whether the design is of the LB type or not.

An Embedding Theorem for Balanced Incomplete Block Designs

From a symmetric balanced incomplete block design we may construct a derived design by deleting a block and its varieties. But a design with the parameters of a derived design may not be embeddable in a symmetric design. Bhattacharya (1) has such an example with λ = 3 . When λ = 1, the derived design is a finite Euclidean plane and this can always be embedded in a corresponding symmetric design which will be a finite projective plane.

The Impossibility of Certain Symmetrical Balanced Incomplete Block Designs

An arrangement of $v$ varieties or treatments in $b$ blocks of size $k, (k < v),$ is known as a balanced incomplete block design if every variety occurs in $r$ blocks and any two varieties occur together in $\lambda$ blocks. These parameters obviously satisfy the equations \begin{equation*}\tag{1} bk = vr\end{equation*}\begin{equation*}\tag{2} \lambda(v - 1) = r(k - 1)\end{equation*} Fisher [1] has also proved that the inequality \begin{equation*}\tag{3} b \geq v, \quad r \geq k\end{equation*} must hold. If $v, b, r, k$ and $\lambda$ are positive integers satisfying (1), (2) and (3), then a balanced incomplete block design with these parameters possibly exists, but the actual existence of a combinatorial solution is not ensured. These conditions are thus necessary but not sufficient for the existence of a design. Fisher and Yates in their tables [2] have listed all designs with $r \leq 10$ and given combinatorial solutions, where known. A balanced incomplete block design in which $b = v$, and hence $r = k$ is called a symmetrical balanced incomplete block design. The impossibility of the symmetrical designs with parameters $v = b = 22, r = k = 7, \lambda = 2$ and $v = b = 29, r = k = 8, \lambda = 2$ was first demonstrated by Hussain [3], [4] essentially by the method of enumeration. The object of the present note is to give an alternative simple proof of the impossibility of these designs and to show that the only unknown remaining symmetrical design in Fisher and Yates' tables, viz. $v = b = 46, r = k = 10, \lambda = 2,$ is definitely impossible. Symmetrical designs with $\lambda \leq 5, r, k \leq 20$, which are impossible combinatorially, are also listed.

Combinatorial Problems

Let it be required to arrange v elements into v sets such that every set contains exactly k distinct elements and such that every pair of sets has exactly elements in common . This combinatorial problem is studied in conjunction with several similar problems, and these problems are proved impossible for an infinitude of v and k. An incidence matrix is associated with each of the combinatorial problems, and the problems are then studied almost entirely in terms of their incidence matrices. The techniques used are similar to those developed by Bruck and Ryser for finite projective planes [3]. The results obtained are of significance in the study of Hadamard matrices [6;8], finite projective planes [9], symmetrical balanced incomplete block designs [2; 5], and difference sets [7].