Here is a very different picture of the Shrikhande graph from that we have seen before (Fig. 7.1).

The graph appears to have 25 vertices. But the purpose of the small single and double arrows is to tell us that the five vertices along the bottom of the parallelogram are to be identified with the five vertices along the top, while the five on the left are identified with the five on the right, keeping the same order in each case. (So for example the four vertices of the parallelogram become a single vertex after this identification.)

Before proceeding, we stop to identify the graph with the Shrikhande graph, using the description given in Section 2.6.

Through the window of the Shrikhande graph, we have introduced you to many important topics in discrete mathematics. We hope that you have been inspired to learn more about some of the topics, so we provide a list of books that go further than we have done. For a textbook on discrete mathematics, somewhat similar in spirit to this book, we recommend van Lint and Wilson [71].

For general graph theory, we suggest the books by Bollobás [18] and Diestel [42].

A series of books edited by Lowell Beineke and Robin Wilson [10, 11, 12] give a good overview of several aspects of graph theory: algebraic, topological, structural and chromatic [13].

In this chapter, we use the classification of the root systems with all roots of the same length to determine the graphs G whose adjacency matrix A(G) has least eigenvalue −2 or larger. This classification includes the Shrikhande graph and gives us another, quite different, proof of Shrikhande’s Theorem.

Note that if G is a graph with at least one edge, then the sum of the eigenvalues of A(G) is zero (the trace of A(G)), and hence G has both positive and negative eigenvalues. So, for non-null graphs, the smallest eigenvalue is negative.

‘Graph theory’ is a relatively recent addition to the mathematical canon. Some date its origins to the Swiss mathematician Leonhard Euler’s (1708–83) work on the bridges of Königsberg (now Kaliningrad) in 1736. The city was at the junction of two rivers forming an island, and seven bridges connected the four regions of the city (Fig. 2.1). According to legend, the citizens asked Euler whether it was possible to take a walk, crossing each of the bridges just once and returning to the starting point. Euler showed that this was not possible.

(The picture shows the layout of the bridges in Euler’s time. Since then, destruction in war and new construction have changed the layout.

Ambat Vijayakumar, an author of this book, writes:

The International Conference on Recent Trends in Graph Theory and Combinatorics (ICRTGC) was held in Cochin, India. It was organized by the Cochin University of Science and Technology, India, during 7–10 June 2010, as a satellite conference of the International Congress of Mathematicians (ICM) 2010 held in Hyderabad, India. The conference logo (Fig. 1.1) was the renowned ‘Shrikhande Graph’. I had only a vague memory of having met S. S. Shrikhande in a conference held at the University of Mumbai and I had never heard about his contributions to combinatorial designs, association schemes, and the Shrikhande Graph itself before 2010. Although I do not wish to find excuses for my ignorance, it is surprising that the Shrikhande Graph came to my mind.

We saw earlier that the Shrikhande graph is a strongly regular graph, with parameters (16,6,2,2). In Chapter 3, we saw that this implies that the adjacency matrix A satisfies the two equations

AJ = 6J, A2 = 4I + 2J,

where I and J are the identity and all-1 matrices of order 16.

The first equation shows that the all-1 vector j is an eigenvector of A with eigenvalue 6. From the theory of real symmetric matrices we know that any other eigenvector v is orthogonal to j, and hence if the corresponding eigenvalue is θ.

In this chapter, we give the various properties and parameters of the Shrikhande graph; these have all been introduced earlier.

Vertices, edges, regularity The Shrikhande graph has 16 vertices and 48 edges. It is regular with degree 6. It is strongly regular: any two vertices have two common neighbours.

Symmetry The Shrikhande graph is vertex, edge and flag transitive. Its automorphism group is (Z4)2 : D12, of order 192, and has two orbits on non-edges.

Euler and Hamilton Since all vertices have even degree, the graph is Eulerian.

It is also Hamiltonian. Consider the description in Section 2.6. By using only four of the six types of edges given there

Let n be a positive integer. A Latin square of order n is an n × n array whose entries are taken from an alphabet of size n, in such a way that each row or column of the array contains each letter in the alphabet precisely once.

The “letters” in the alphabet could be letters, numbers, colours or indeed any distinguishable symbols. Figure 6.1 gives two examples.

As far as we know, the first person to study Latin squares was the Korean mathematician Choi Seok-jeong (1646–1715), in his book Gusuryak (Fig. 6.2). The name was given by the Swiss mathematician Leonhard Euler (1707–83); we will see why he chose this name later.

Root systems are beautiful geometric objects in Euclidean space, which crop up in many parts of mathematics, including Lie algebras, singularity theory, mathematical physics and graph theory.

In this chapter, we will use graph theory to discuss the famous ADE classification of root systems in which all roots have the same length. This will then be used to determine the graphs whose adjacency matrix has least eigenvalue −2 or greater in Chapter 9, where we will see a connection between the Shrikhande graph and exceptional root systems.

We work in the Euclidean space V = Rd, with the standard inner product.

Given a non-zero vector u ∈ V, there is a unique hyperplane Hu through the origin which is perpendicular to u.We define the reflection ru in this hyperplane to be the linear map which fixes every vector in Hu and maps u to −u.

In this chapter we examine other areas in which the Shrikhande graph has a role to play, including Seidel switching and equiangular line sets (which give rise to our final construction of the Shrikhande graph), design theory, Hadamard matrices and distance-regular graphs.

We begin with a short section indicating a few directions in which the study of the Shrikhande graph has been taken. Most of the detail is omitted, and we refer to the cited papers.