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.