In our earlier chapters, we have dealt somewhat at length with the questions of extensions and embeddings of (symmetric) designs, quasi-symmetric designs, and strongly regular graphs. The most elegant examples of these situations are provided by the Witt designs. Historically, statisticians were the first to make a systematic and exhaustive study of block designs, particularly from the point of view of constructions. However, block designs with regular and nice structural properties are generally obtained by making themselves available from groups. It is from this point of view that the (study of) Witt designs becomes extremely important. At present, there are at least two objects (excluding objects such as the Leech lattice) that are equivalent to Witt designs. These are the Golay codes (from coding theory) and the five Mathieu groups, all of which were the first examples of sporadic simple groups that have been completely determined now.

From the combinatorialist's point of view, a substantial portion of the research work in design theory centers around various characterizations of Witt designs by their properties. In this connection, also note that a result of Ito et al. and Bremner implies that the only non-trivial quasi-symmetric 4-design is the Witt 4-design or its complement. It has been conjectured that the only nontrivial quasi-symmetric 3-designs (other than the Hadamard 3-designs) are the 3-designs related to the Witt designs or their complements (see Chapter IX).

In our earlier chapters, we have looked at various classes of (sometimes only parametrically possible) quasi-symmetric designs. This chapter is devoted to the study of quasi-symmetric designs in general, particularly from a structural point of view. While Cameron's Theorem (Theorem 1.29) certainly boosted the interest in quasi-symmetric designs, it seems to be only in the last ten years or so that the structural investigations of quasi-symmetric designs began. The investigations are far from complete and we wish to give an account of the work in this area. The strong regularity of the block graph of a q.s. (= quasi-symmetric) design has been exploited in a paper of Neumaier. However, the design-structural properties of q.s. designs were probably first studied in a paper of Baartmans and M.S. Shrikhande. In that paper, q.s. designs with (x, y) = (0, y), y ≥ 2 and with no three mutually disjoint blocks were studied. Typical examples are Eλ and Qλ of Chapter VII (Convention 7.12). Various modifications and improvements of the results in that paper have been obtained and the results have been generalized in two different directions. The main part of this chapter is to give a summary of most of the results in these directions.

Throughout this chapter, D denotes a quasi-symmetric (q.s.) design with parameter set (v, b, r, k, λ) and with block intersection numbers x and y where x ≤ y.

The preceding chapters were focused on the mathematical foundations of infinite electrical networks, existence and uniqueness theorems being their principal result. Generality was a concomitant aim of those discussions. For the remaining chapters, we shift our attention to particular kinds of networks (namely, the infinite cascades and grids) that are more closely related to physical phenomena. Two examples of this significance were given in Section 1.7, and more will be discussed in Chapter 8. Our proofs will now be constructive, and consequently methods for finding voltage-current regimes will be encompassed. Moreover, various properties of voltage-current regimes will examined.

We must now be specific about any network we hope to analyze. In particular, its graph and element values need to be stipulated everywhere. An easy way of doing this is to impose some regularity upon the network. Most of this chapter (Sections 6.1 to 6.8) is devoted to the simplest of such regularities, the periodic two-times chainlike structures. They are 0-networks appearing in two forms. One form will be called a one-ended grounded cascade and is illustrated in Figure 6.1; it has an infinite node as one of its spines, and a one-ended 0-path as the other spine. The other form will be called a one-ended ungrounded cascade and is shown in Figure 6.2; in this case, both spines are one-ended 0-paths. The third possibility of both spines being infinite nodes is a degenerate case and will not be discussed.

The theory of electrical networks became fully launched, it seems fair to say, when Gustav Kirchhoff published his voltage and current laws in 1847 [72]. Since then, a massive literature on electrical networks has accumulated, but almost all of it is devoted to finite networks. Infinite networks received scant attention, and what they did receive was devoted primarily to ladders, grids, and other infinite networks having periodic graphs and uniform element values. Only during the past two decades has a general theory for infinite electrical networks with unrestricted graphs and variable element values been developing. The simpler case of purely resistive networks possesses the larger body of results. Nonetheless, much has also been achieved with regard to RLC networks. Enough now exists in the research literature to warrant a book that gathers the salient features of the subject into a coherent exposition.

As might well be expected, the jump in complexity from finite electrical networks to infinite ones is comparable to the jump in complexity from finite-dimensional spaces to infinite-dimensional spaces. Many of the questions we conventionally ask and answer about finite networks are unanswerable for infinite networks – at least at the present time.

As was indicated in Example 1.6-3, the total power dissipated in the resistances by a voltage-current regime, satisfying Ohm's law, Kirchhoff's current law at finite nodes, and Kirchhoff's voltage law around finite loops, need not be finite. Moreover, these laws need not by themselves determine the regime uniquely. However, if voltage-current pairs are assigned to certain branches, the infinite-power regime may become uniquely determined. The latter result requires in addition the “nonbalancing” of various subnetworks, as is explained in the next section. In which branches the voltage-current pairs can be arbitrarily chosen and how the nonbalancing criterion can be specified are the issues resolved in this chapter. The discussion is based on a graph-theoretic decomposition of the countably infinite network into a chainlike structure, which was first discovered by Halin for locally finite graphs [63]. That result has been extended to graphs having infinite nodes [166]. The chainlike structure implies a partitioning of the network into a sequence of finite subnetworks, which can be analyzed recursively to determine the voltage-current pair for every branch. We call this a limb analysis.

As was mentioned before, in most of this book we restrict our attention to resistive networks. However, a limb analysis can just as readily embrace complex-valued voltages, currents, and branch parameters. In short, a limb analysis can be used for a phasor representation of an AC regime or for the complex representation of a Laplace-transformed transient regime in a linear RLC network [166].

If D is a quasi-symmetric design, then we know from Chapter III that its block graph is strongly regular. We have also seen that by imposing some extra structure on the block graph we can pin down the design D. To a quasi-symmetric design, we can associate some other naturally associated strongly regular graphs. Conversely, starting from a graph we may sometimes produce a design. If the graph г has some further structure, then in some cases the associated designs may have additional properties. We also enlarge the class of incidence structures under consideration and study graphs associated with partially balanced incomplete block designs with a two class association scheme. Partially balanced incomplete block designs are simply 1-designs with occurrences of point-pairs in blocks determined by adjacencies in the superposed strongly regular graph.

A restricted class of the partially balanced incomplete block designs (PBIBDs) is the class of special partially balanced incomplete block designs (SPBIBDs) which, in addition, satisfy a point-block regularity condition. A generalization of the notion of an SPBIBD is the notion of a partial geometric design of Bose, S.S. Shrikhande, and Singhi. The former structures were introduced by Bridges and M.S. Shrikhande in order to obtain a unified approach to many graph embedding problems considered earlier. A nice outcome of the linear algebraic techniques is the spectral characterization of partial geometric designs by Bose, Bridges, and M.S. Shrikhande.

Perhaps the most important infinite electrical networks with respect to physical phenomena – putting aside the finite networks – are the infinite grids. This is because finite-difference approximations of various partial differential equations have realizations as electrical networks whose nodes are located at the sample points of the approximation. Those sample points are distributed in accordance with increments in each of the coordinates, hence the gridlike structure. Moreover, if the phenomenon exists throughout an infinite domain, it is natural (but, to be sure, not always necessary) to choose an infinity of sample points. In this way, one is led to infinite electrical grids as models for the so-called “exterior problems” of certain partial differential equations. Two cases of this were presented in Section 1.7, and more will be discussed in the next chapter. The grids we examine are of two general types: the grounded grids, wherein a resistor connects each node to a common ground node, and the ungrounded grids, wherein those grounding resistors are entirely absent. Grounded grids are readily analyzed, but ungrounded grids are more problematic because of a singularity in a certain function that characterizes the network.

This chapter is devoted to rectangular grids, the natural finite difference model for Cartesian coordinates, but the analysis can be extended to other coordinate systems such as cylindrical and spherical ones [180], [183], [184], [188].

Our aim in this chapter is to gather together some basic results from strongly regular graphs and partial geometries. These topics have had a profound influence in the area of combinatorial designs after Bose's classical paper of 1963. The results of the first two chapters will provide the necessary background for later chapters. We refer to Harary for the necessary background in graph theory. Marcus and Mine will generally suffice for details of matrix results used. For further applications of matrix tools in a variety of problems on designs, we refer to M.S. Shrikhande.

Let Γ be a finite undirected graph on n vertices. The adjacency matrix A of Γ is a square matrix of size n. The diagonal entries of A are zero and for i ≠ j, the (i, j) entry of A is 1 or 0 according as the vertices i and j are joined by an edge or not. There are other types of adjacency matrices used. For example in Seidel, or Goethals and Seidel, a (0, ± 1) adjacency matrix is used.

A graph Γ is called regular of valency a if A has constant row sum a. The adjacency matrix reflects many other graphical properties of Γ.

We now state two basic definitions. Firstly, a matrix A is permutationally congruent to a matrix B if there is a permutation matrix P such that A = PtBP.

All through the development of this monograph up to the present point (especially Chapters V through VIII), we have looked at various properties and characterizations, both parametric and geometric, of quasi-symmetric designs, essentially to the point of convincing ourselves that the problem of determination of all the quasi-symmetric designs is indeed a hard problem. With that background, this chapter will show us that the situation for quasi-symmetric 3-designs is more promising. This is to be expected since a derived design of a q.s. 3-design at any point is also quasi-symmetric and that gives us more information (Example 5.24). However, quite unlike the case of q.s. 2-designs, very few q.s. 3-designs seem to be known. In fact, up to complementation there are only three known examples of q.s. 3-designs with x ≠ 0. Application of the ‘polynomial method’ to this rather mysterious situation (see Cameron) is the theme of the present chapter.

Cameron's Theorem (Theorem 1.29) has been one of the focal points of our study of q.s. designs. This is more so for q.s. 3-designs since the extensions of symmetric designs obtained in that theorem are in fact quasi-symmetric with x = 0 (and conversely). Despite its completeness, the classification given by Cameron's theorem is perhaps and unfortunately only parametric as revealed in the following discussion. A Hadamard 3-design exists if and only if a Hadamard matrix of the corresponding order exists. Hadamard matrices are conjectured to exist for every conceivable (i.e., a multiple of four) order.

In this first chapter, we collect together and review some basic definitions, notation, and results from design theory. All of these are needed later on. Further details or proofs not given here may be found, for example, in Beth, Jungnickel and Lenz, Dembowski, Hall, Hughes and Piper, or Wallis. We mention also the monographs of Cameron and van Lint, Biggs and White, and the very recent one by Tonchev.

Let X = {x1,x2,…,xv} be a finite set of elements called points or treatments and β = { B1,B2,…,Bb} be a finite family of distinct k-subsets of X called blocks. Then the pair D = (X, β) is called a t-(v, k, λ) design if every t-subset of X occurs in exactly λ blocks. The integers v, k, and λ are called the parameters of the t-design D. The family consisting of all k-subsets of X forms a k-(v, k, 1) design which is called a complete design. The trivial design is the v-(v, v, 1) design. In order to exclude these degenerate cases we assume always that v > k >t ≥ 1 and λ ≥ 1. We use the term finite incidence structure to denote a pair (X, β), where X is a finite set and β is a finite family of not necessarily distinct subsets of X. In most of the situations of interest in the later chapters, however, we will have to tighten these restrictions further.

The adjective “nonlinear” will be used inclusively by taking “linear” to be a special case of “nonlinear.” As promised, we present in this chapter two different theories for nonlinear infinite networks. The first one is due to Dolezal and is very general in scope – except that it is restricted to 0-networks. It is an infinite-dimensional extension of the fundamental theory for scalar, finite, linear networks [67], [115], [127]. In particular, it examines nonlinear operator networks, whose voltages and currents are members of a Hilbert space ℋ; in fact, infinite networks whose parameters can be nonlinear, multivalued mappings restricted perhaps to subsets of ℋ are encompassed. As a result, virtually all the different kinds of parameters encountered in circuit theory – resistors, inductors, capacitors, gyrators, transformers, diodes, transistors, and so forth – are allowed. However, there is a price to be paid for such generality: Its existence and uniqueness theorems are more conceptual than applicable, because their hypotheses may not be verifiable for particular infinite networks. (In the absence of coupling between branches, the theory is easy enough to apply; see Corollary 4.1-7 below.) Nonetheless, with regard to the kinds of parameters encompassed, this is the most powerful theory of infinite networks presently available. Dolezal has given a thorough exposition of it in his two books [40], [41]. However, since no book on infinite electrical networks would be complete without some coverage of Doleza's work, we shall present a simplified version of his theory.

The purposes of this initial chapter are to present some basic definitions about infinite electrical networks, to show by examples that their behaviors can be quite different from that of finite networks, and to indicate how they approximately represent various partial differential equations in infinite domains. Finally, we explain how the transient responses of linear RLC networks can be derived from the theory of purely resistive networks; this is of interest because most of the results of this book are established in the context of resistive networks.

Notations and Terminology

Let us start by reviewing some symbols and phraseology so as to dispel possible ambiguities in our subsequent discussions. We follow customary usage; hence, this section may be skipped and referred to only if the need arises. Also, an Index of Symbols is appended for the more commonly occurring notations in this book; it cites the pages on which they are defined.

Let X be a set. X is called denumerably infinite or just denumerable if its members can be placed in a one-to-one correspondence with all the natural numbers: 0, 1, 2,. … X is called countable if it is either finite or denumerable. In this book the set of branches of any network will always be countable.

The notation {x ∈ X: P(x)}, or simply {x: P(x)} if X is understood, denotes the set of all x ∈ X for which the proposition P(x) concerning x is true.

An important class of quasi-symmetric designs is the class of symmetric designs characterized among the former larger class by the property of having only one block intersection number. Though the symmetric designs by themselves are improper quasi-symmetric designs, as we already saw in Theorem 1.29, the extendable symmetric designs open up many possibilities for the parameter sets of proper quasi-symmetric designs. In fact, the classification theorem of Cameron (Theorem 1.29) has given rise to a considerable activity in the area of quasi-symmetric 2 and 3-designs. We will choose to postpone these topics to later chapters and concentrate here on the structure of those 3-designs that can be obtained as extensions of symmetric designs. In doing so, we will also consider some other quasi-symmetric designs (such as a residual design) associated with the extension process.

Recalling Cameron's Theorem, observe that it classifies extendable symmetric designs into four sets: an infinite set of symmetric designs the first object of which is a projective plane of order four, an infinite set of all the Hadamard 2-designs, a projective plane of order ten and a symmetric (495, 39, 3)-design. The existence of a Hadamard 2-design is equivalent to the existence of the Hadamard matrix of corresponding order. Nothing is known about a (495, 39, 3)-design. In this chapter, we first consider the extension question of a projective plane of order ten.

The importance of coding theory as a valuable tool in the study of designs has been known for quite some time. We mention, for example, M. Hall, Jr., MacWilliams and Sloane, Pless, and also the monographs by Cameron and van Lint and Tonchev. Recently Tonchev, Calderbank, and Bagchi have proved some very nice results about designs using coding theory. We have referred to Bagchi's result (Theorem 7.30) in an earlier chapter.

The paper of Tonchev has shown the link between quasi-symmetric designs and self-dual codes. Calderbank, has proved some elegant non-existence criteria about 2-designs in terms of their intersection numbers. The proof of one of Calderbank's results depends on some deep theorems of Gleason and Mallows, and MacWilliams-Sloane on weight enumerators of certain self-dual codes. The results of Calderbank, and Tonchev when specialized to quasi-symmetric designs give strong results about existence, non-existence or uniqueness. For example, Tonchev shows the falsity of a part of the well known Hamada conjecture concerning the rank of the incidence matrix of certain 2-designs. Some results of Tonchev and Calderbank, seem to have been motivated by Neumaier's table of exceptional quasi-symmetric designs given in Chapter VIII.

The purpose of this chapter is to review some of the results of Tonchev, and Calderbank, which rely on codes as one of their principal tools.