THE SHRIKHANDE GRAPH. 1. Introduction

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1 THE SHRIKHANDE GRAPH RYAN M. PEDERSEN Abstract. In 959 S.S. Shrikhande wrote a paper concerning L 2 association schemes []. Out of this paper arose a strongly regular graph with parameters (6, 6, 2, 2) that was not isomorphic to L 2 (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.. Introduction A well studied and simple family of strongly regular graphs are called the square lattice graphs L 2 (n). These graphs have parameters (n 2, 2(n ), 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 (6, 6, 2, 2). The non-lattice graph with these parameters is known as the Shrikhande graph. In what follows, we consider various constructions of this graph, along with a discussion of its various properties. 2. Constructions 2.. The Original Construction. We begin by describing what is done in the original paper []. 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 6, 27.

2 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 =, 2) then they occur together in λ i 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 L 2 (s) where v = s 2, then the design is called an L 2 association scheme. Now In [] the following is shown. Theorem. If the parameters of the second kind for a partially balanced incomplete block design with s 2 treatments with two associate classes are given by n = 2s 2, p = s 2, p2 = 2, then the design has L 2 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 = 6, n = 6, n 2 = 9 p = 2, p 2 = 3, p 22 = 6 p 2 = 2, p 2 2 = 4, p 2 22 = 4. One of these is of course given by the graph L 2 (4). The other, however, is the Shrikhande graph The Switching Construction. A more standard way of constructing the Shrikhande graph is given in [5] as follows. Start with L 2 (4) which is shown in figure.

3 TOPICS PAPER, INSTRUCTOR WILLIAM CHEROWITZO 3 Figure. L 2 (4) Figure 2. The Shrikhande graph If we then perform the operation of switching with respect to the vertices 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 The Code Graph Construction. Another construction uses the concept of a code graph defined here.

4 4 RYAN M. PEDERSEN Figure 3. Codeword construction Definition. A code graph Γ(C) is a graph whose vertices are codewords 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,,,, and those obtained by a cyclic permutation of the six entries, then one obtains the Shrikhande graph. This is illustrated in figure 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 n 2 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 (n 2, 3(n ), 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

5 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 (6, 9, 4, 6). The complements of these graphs are the graphs L 2 (4), and the Shrikhande graph. See [4] for a deeper look at this construction The Cayley Graph Construction. Related to the previous construction, our final construction is found in [8]. We need a couple 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 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 S. Now consider the group H = Z 4 Z 4 with S = {±(, ), ±(, ), ±(, )}. 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 6 as well: a, b a 8, b 2, baba 3 with S = {a 3, a 5, a 5 b, a 7 b, a 2 b, a 6 b}, a, b a 8, b 2, baba 5 with S = {a 3, a 5, b, a 6 b, a 3 b, a 7 b}, a, b, c a 4, b 2, c 2, [a, b], [b, c], (ca) 2 with S = {ab, a 3 b, abc, bc, a 2 c, ac}. 3. Properties 3.. General Properties. We begin by listing the general properties that this graph has. Most (but not all) of these were taken from [2]. We list these here without proof.

6 6 RYAN M. PEDERSEN It is a (, 2) graph. It is locally hexagon. It has an automorphism of order 92 that acts sharply transitive on ordered triangles. Both the independence and chromatic number are 4. The complement (which is a Latin square graph) has independence 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]) Special Properties. The Shrikhande graph arises in several different 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 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.

7 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 L 2 (4) is minimal, but both these graphs share the same set of parameters, and are non-isomorphic. Finally, as shown in [] take the cross product of a K 2 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 Γ i (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 parameters (v, k, λ, µ) whenever Γ is edge-regular and Γ (a) Γ (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 Γ 3 (a) is the Shrikhande graph. Then we construct the following design (P, B) by: P B = Γ 3 (a) = a Shrikhande graph, = Γ 2 (a), and p P is incidence with B B whenever p and B are adjacent in Γ. This forms a 2 (6, 4, 8) design with b = 36, and r = 9.

8 8 RYAN M. PEDERSEN References [] Abdelhafid Berrachedi, Ivan Havel, and Henry Martyn Mulder. Spherical and clockwise spherical graphs. Czechoslovak Mathematical Journal, 53(28):295 39, 23. [2] A.E. Brouwer. Shrikhande graph. aeb/drg/graphs/shrikhande.html, 27. [3] A.E. Brouwer, A.M. Cohen, and A. Neumaier. Distance-Regular Graphs. Springer-Verlag, Berlin Heidelberg, 989. [4] Peter Cameron. Strongly regular graphs [5] P.J. Cameron and J.H. Van Lint. Designs, Graphs, Codes and their Links. Cambridge University Press, 4 West 2th Street, New York, NY -42, 99. [6] A.L. Gavrilyuk and A.A. Makhnev. Amply regular graphs and block designs. Siberian Mathematical Journal, 47(4):62 633, 26. [7] A.A. Makhnev and D.V. Paduchikh. Locally shrikhande graphs and their automorphisms. Siberian Mathematical Journal, 39: , 998. [8] Stefko Miklavic and Primoz Potocnik. Distance-regular circulants. European Journal of Combinatorics, 24: , 23. [9] Rene Peeters. Uniqueness of strongly regular graphs having minimal p-rank. Linear Algebra and its Applications, 226(228):9 3, 995. [] Rene Peeters. On the p-ranks of the adjacency matrices of distance-regular graphs. Journal of Algebraic Combinatorics, 5:27 49, 22. [] S.S. Shrikhande. The uniqueness of the L 2 association scheme. The Annals of Mathematical Statistics, 3(3):78 798, 959. [2] Wolfram. Shrikhande graph