THE SHRIKHANDE GRAPH
RYAN M. PEDERSEN
Abstract. In 1959 S.S. Shrikhande wrote a paper concerning L2 association schemes [11]. Out of this paper arose a strongly reg- ular graph with parameters (16, 6, 2, 2) that was not isomorphic
to L2(4). This graph turned out to be important in the study of strongly regular graphs as a whole. In this paper, we survey the various constructions and properties of this graph.
1. Introduction
A well studied and simple family of strongly regular graphs are called the square lattice graphs L2(n). These graphs have parame- ters (n2, 2(n − 1), n − 2, 2). Now strongly regular graphs with these parameters are unique for all n except n = 4. However, when n = 4 we have two non-isomorphic strongly regular graphs with parameters (16, 6, 2, 2). The non-lattice graph with these parameters is known as the Shrikhande graph. In what follows, we consider various construc- tions of this graph, along with a discussion of its various properties.
2. Constructions
2.1. The Original Construction. We begin by describing what is done in the original paper [11]. Note first that a strongly regular graph is equivalent to a two-class association scheme. Now if we can arrange the v vertices (points) into b subsets (blocks) such that
Date: November 16, 2007. 1 2 RYAN M. PEDERSEN
(a) Each block contains k points (all different), (b) Each point is contained in r blocks, (c) if any two points are ith associates (i = 1, 2) then they occur
together in λi blocks, then we call this design D a partially balanced incomplete block (PBIB) design . Now if the PBIB design comes from the strongly regular graph 2 L2(s) where v = s , then the design is called an L2 association scheme. Now In [11] the following is shown.
Theorem 1. If the parameters of the second kind for a partially bal- anced incomplete block design with s2 treatments with two associate classes are given by
1 2 n1 =2s − 2, p11 = s − 2, p11 =2, then the design has L2 association scheme if s =2, 3, or s> 4.
However when s = 4 there exists two non-isomorphic PBIB designs with the following parameters
v = 16, n1 = 6, n2 = 9 1 1 1 p11 = 2, p12 = 3, p22 = 6 2 2 2 p11 = 2, p12 = 4, p22 = 4.
One of these is of course given by the graph L2(4). The other, however, is the Shrikhande graph.
2.2. The Switching Construction. A more standard way of con- structing the Shrikhande graph is given in [5] as follows. Start with
L2(4) which is shown in figure 1. TOPICS PAPER, INSTRUCTOR WILLIAM CHEROWITZO 3
Figure 1. L2(4)