Combinatorics is generally concerned with counting arrangements within a finite set. One of the basic problems is to determine the number of possible configurations of a given kind. Even when the rules specifying the configuration are relatively simple, the questions of existence and enumeration often present great difficulties. Besides counting, combinatorics is also concerned with questions involving symmetries, regularity properties, and morphisms of these arrangements. The theory of block designs is an important area where these facts are very apparent. The study of block designs combines number theory, abstract algebra, geometry, and many other mathematical tools including intuition. In the words of G.C. Rota (in: Studies in Combinatorics, Mathematical Association of America, 1978),

“Block designs are generally acknowledged to be the most complex mathematical structures that can be defined from scratch in a few lines. Progress in understanding and classification has been slow and proceeded by leaps and bounds, one ray of sunlight followed by years of darkness. …This field has been enriched and made even more mysterious, a battleground of number theory, projective geometry and plain cleverness. This is probably the most difficult combinatorics going on today…”

In the last few years, some new text-books (Beth, Jungnickel and Lenz, Hughes and Piper, Wallis) on Design Theory have been published. Dembowski's ‘Finite Geometries’, M. Hall Jr.'s ‘Combinatorial Theory,’ and Ryser's ‘Combinatorial Mathematics’ are regarded as some of the classic references in combinatorics, particularly in the area of designs.

Suppose D is a t-(v, k, λ) design with blocks B1, B2,…, Bb. The cardinality | Bj∩Bj |, i ≠ j, is called an intersection number of D. Assume that x1, x2,…, xs are the distinct intersection numbers of the design D. Specifying some of the xi's or the number s can sometimes provide very useful information about the design. For instance, any 2-design with exactly one intersection number must necessarily be symmetric. Any 2-design with exactly the two intersection numbers 0 and 1 must be a non-symmetric 2-(v, k, 1) design.

In this chapter, we discuss designs which are in a sense “close” to symmetric designs. These are t-(v, k, λ) designs with exactly two intersection numbers. Such designs are called quasi-symmetric. We believe this concept goes back to S.S. Shrikhande who considered duals of designs with λ = 1. We let x, y stand for the intersection numbers of a quasi-symmetric design with the standard convention that x < y.

Before proceeding further, we list below some well known examples of quasi-symmetric designs.

Example 3.1. Let D be a multiple of a symmetric 2-(v, k, λ) design. Then D is a quasi-symmetric 2-design with x = λ and y = k.

Example 3.2. Let D be a 2-(v, k, 1) design with b > v. Then obviously D is quasi-symmetric with x = 0 and y = 1.

In the previous Chapter IV, we looked at the possibility of a strongly regular graph G containing vertex subsets which define a design or a PBIBD when the adjacency is suitably translated into incidence. A fairly general set-up was given in the notion of an SPBIBD of the last chapter. Essentially, the block graphs of interesting incidence structures are, of course, strongly regular but the combination of relations between points, blocks and point-blocks sometimes produces larger strongly regular graphs, and the procedure is often reversible. In a recent paper Haemers and Higman obtained an elegant combinatorial generalization of these ideas, where a graph is partitioned into two strongly regular graphs but these graphs are not necessarily assumed to arise out of a design or a PBIBD. The present chapter is almost entirely based on the paper of Haemers and Higman. Here we consider a strongly regular graph г0 whose vertex set V0 can be written as a disjoint union of two sets V1 and V2 such that the induced graph гi on Vi, i = 1, 2 has some nice properties. These nice properties include regularity, strong regularity, being a complete graph or its complement, etc.

Definition 5.1. Let m < n and let ρ1 ≥ ρ2 ≥ … ≥ ρn and σ1 ≥ σ2 ≥ … ≥ σm be two sequences of real numbers.

In this last chapter we shall survey a number of instances where infinite electrical networks are useful models of physical phenomena, or serve as analogs in some other mathematical disciplines, or are realizations of certain abstract entities. We shall simply describe those applications without presenting a detailed exposition. To do the latter would carry us too far afield into quite a variety of subjects. However, we do provide references to the literature wherein the described applications can be examined more closely.

Several examples are presented in Sections 8.1 and 8.2 that demonstrate how the theory of infinite electrical networks is helpful for finding numerical solutions of some partial differential equations when the phenomenon being studied extends over an infinite region. The basic analytical tool is an operator version of Norton's representation, which is appropriate for an infinite grid that is being observed along a boundary. In effect, the infinite grid is replaced by a set of terminating resistors and possibly equivalent sources connected to the boundary nodes. In this way, the infinite domain of the original problem can be reduced to a finite one – at least so far as one of the spatial dimensions is concerned. This can save computer time and memory-storage requirements.

In Section 8.3 we describe two classical problems in the theory of random walks on infinite graphs and state how infinite-electrical-network theory solves those problems. Indeed, resistive networks are analogs for random walks.

In 1971 Harley Flanders [51] opened a door by showing how a unique, finite-power, voltage-current regime could be shown to exist in an infinite resistive network whose graph need not have a regular pattern. To be sure, infinite networks had been examined at least intermittently from the earliest days of circuit theory, but those prior works were restricted to simple networks of various sorts, such as ladders and grids. For example, infinite uniform ladder networks were analyzed in [31], [73], and [139], works that appeared 70 to 80 years ago. Early examinations of uniform grids and the discrete harmonic operators they generate can be found in [35], [44], [45], [47], [52], [54], [65], [84], [126], [135], [143].

Flanders' theorem, an exposition of which starts this chapter, is restricted to locally finite networks with a finite number of sources. Another tacit assumption in his theory is that only open-circuits appear at the infinite extremities of the network. The removal of these restrictions, other extensions, and a variety of ramifications [158], [159], [163], [177] comprise the rest of this chapter. However, the assumptions that the network consists only of linear resistors and independent sources and is in a finite-power regime are maintained throughout this chapter.

Actually, finite-power theories for nonlinear networks are now available, and they apply just as well to linear networks as special cases. One is due to Dolezal [40], [41], and the other to DeMichelle and Soardi [37].

We have already seen that the idea of “connections at infinity” can be encompassed within electrical network theory through the invention of 1-nodes. A consequence is the genesis of transfinitely connected graphs, that is, graphs having pairs of nodes that are connected through transfinite paths but not through finite ones. An example of this is provided by Figure 3.4; there is no finite path connecting a node of branch a to a node of branch α, but there are 1-paths that do so. The maximal, finitely connected subnetworks of that figure are the 0-sections of the 1-graph, and the 1-nodes described in Example 3.2-5 connect those infinitely many 0-sections into a 1-graph.

There is an incipient inductive process arising here. Just as 0-nodes connect branches together to produce a 0-graph, so too do 1-nodes connect 0-sections together to produce 1-graphs. The purpose of the present chapter is to pursue this induction. While doing so, we will discover an infinite hierarchy of transfinite graphs. It turns out that most of the electrical network theory discussed so far can be transferred to such graphs to obtain an infinite hierarchy of transfinite networks. Thus, an electrical parameter in one branch can affect the voltage-current pair of another branch, even when every path connecting the two branches must pass through an “infinity of infinite extremities.”

To think about this in another way, let each 0-section of Figure 3.4 be replaced by replicates of the entire 1-graph of that figure.

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].