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THE SHRIKHANDE GRAPH 1. Introduction a Well Studied And 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) Figure 2. The Shrikhande graph If we then perform the operation of switching with respect to the ver- tices on the diagonal, then we obtain the Shrikhande graph as desired. Figure 2 gives a drawing of the graph in its representation on a torus. 2.3. The Code Graph Construction. Another construction uses the concept of a code graph defined here. 4 RYAN M. PEDERSEN 000000 110000 110110 000110 001100 011000 111010 101110 101101 011101 011011 101011 100001 110101 010111 000011 Figure 3. Codeword construction Definition 1. A code graph Γ(C) is a graph whose vertices are code- words of a binary code C with two vertices being adjacent when the codewords differ in two entries. Now if one considers the binary code given by the words 000000, 110000, 010111, 011011, and those obtained by a cyclic permutation of the six entries, then one obtains the Shrikhande graph. This is illustrated in figure 3. 2.4. The Latin Square Graph Construction. Next we consider the concept of a Latin square graph. Definition 2. Suppose we have a Latin square L of order n. Construct a graph Γ as follows: Let the vertices of Γ be the n2 entries of L. Let two vertices be adjacent if and only if the the entries are in the same row, column, or contain the same symbol. Then Γ is called a Latin square graph. Γ is strongly regular with parameters (n2, 3(n − 1), n, 6). Now there are only two non-isomorphic Latin squares of order 4. These are the Caley tables for the Klein group, and the cyclic group TOPICS PAPER, INSTRUCTOR WILLIAM CHEROWITZO 5 of order 4. The Latin square graphs that correspond to these groups are two non-isomorphic strongly regular graphs with parameters (16, 9, 4, 6). The complements of these graphs are the graphs L2(4), and the Shrikhande graph. See [4] for a deeper look at this construction. 2.5. The Cayley Graph Construction. Related to the previous construction, our final construction is found in [8]. We need a cou- ple definitions to begin. Definition 3. Let H be a finite group. A Caley subset S of H is a generating set of H with the property that s ∈ S whenever s−1 ∈ S. Definition 4. A Cayley graph of a group H with respect to S, where S is a Caley subset of H, denoted by Cay(H; S), is the graph with vertex-set H, where x ∈ H is adjacent to y ∈ H whenever xy−1 ∈ S. Now consider the group H = Z4 × Z4 with S = {±(0, 1), ±(1, 1), ±(1, 0)}. Then the graph Cay(H; S) in this case is the Shrikhande graph. We can actually use any of the following three non-Abelian groups of order 16 as well: • ha, b | a8, b2, baba3i with S = {a3, a5, a5b, a7b, a2b, a6b}, • ha, b | a8, b2, baba5i with S = {a3, a5, b, a6b, a3b, a7b}, • ha, b, c | a4, b2,c2, [a, b], [b, c], (ca)2i with S = {ab, a3b, abc, bc, a2c,ac}. 3. Properties 3.1. General Properties. We begin by listing the general properties that this graph has. Most (but not all) of these were taken from [12]. We list these here without proof. 6 RYAN M. PEDERSEN • It is a (0, 2) graph. • It is locally hexagon. • It has an automorphism of order 192 that acts sharply transitive on ordered triangles. • Both the independence and chromatic number are 4. • The complement (which is a Latin square graph) has indepen- dence number 3, and chromatic number 6. • It has edge chromatic number 6. • It has girth 3. • The graph is non-planar. • The characteristic polynomial is (x − 6)(x − 2)6(x + 2)8 We can see from the characteristic polynomial, that the Shrikhande graph has -2 as an eigenvalue. Therefore it is what is known as a Seidel graph. Seidel graphs are a special class of strongly regular graphs and are well studied (see [5]). 3.2. Special Properties. The Shrikhande graph arises in several dif- ferent settings as an exceptional graph. We now discuss some of these types of properties here. We begin with a definition. Definition 5. A group of automorphisms of a connected graph Γ of diameter d is said to be distance-transitive on Γ if it is transitive (on the vertex set of Γ and) on each of the sets {(γ, δ)|d(γ, δ) = i} for 0 ≤ i ≤ d. A graph is called distance-transitive if it is connected and admits a distance-transitive group of automorphisms. It is shown in [3] that a distance-transitive graph is distance-regular. However the converse is not true. In fact, the smallest distance-regular graph that is not distance transitive is the Shrikhande graph. This is perhaps the most popular of the special properties listed in this section. TOPICS PAPER, INSTRUCTOR WILLIAM CHEROWITZO 7 The Shrikhande graph also provides an example of a strongly regular graph with minimal p-rank that is not completely determined by its parameters [9]. In particular, the 2 rank of the Shrikhande graph, and that of L2(4) is minimal, but both these graphs share the same set of parameters, and are non-isomorphic. Finally, as shown in [1] take the cross product of a K2 with the Shrikhande graph. This gives us an example of a weakly spherical graph that is not interval monotone. 4. Connections With Designs We conclude this paper by giving the construction of a design that has the Shrikhande graph as the point graph of the design [6]. Let Γ be a graph, and let a be a vertex of Γ. Define Γi(a) as the sub-graph induced by Γ on the set of all vertices distance i from a. Definition 6. A graph Γ is called an amply regular graph with param- eters (v,k,λ,µ) whenever Γ is edge-regular and Γ1(a) ∩ Γ1(b) contains µ vertices for each pair a, b of vertices at distance 2 in Γ. An amply regular graph is strongly regular when it has diameter 2. Now suppose we take an amply regular graph Γ with diameter 3 with the property that there exists a vertex a such that Γ3(a) is the Shrikhande graph. Then we construct the following design (P, B) by: P = Γ3(a) = a Shrikhande graph, B = Γ2(a), and p ∈ P is incidence with B ∈ B whenever p and B are adjacent in Γ. This forms a 2 − (16, 4, 18) design with b = 360, and r = 90. 8 RYAN M. PEDERSEN References [1] Abdelhafid Berrachedi, Ivan Havel, and Henry Martyn Mulder. Spherical and clockwise spherical graphs. Czechoslovak Mathematical Journal, 53(128):295 – 309, 2003. [2] A.E. Brouwer. Shrikhande graph . http://www.win.tue.nl/∼aeb/drg/graphs/Shrikhande.html, 2007.
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