Switched symplectic graphs and their 2-ranks

We apply Godsil–McKay switching to the symplectic graphs over F2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_2$$\end{document} with at least 63 vertices and prove that the 2-rank of (the adjacency matrix of) the graph increases after switching. This shows that the switched graph is a new strongly regular graph with parameters (22ν-1,22ν-1,22ν-2,22ν-2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2^{2\nu }-1, 2^{2\nu -1}, 2^{2\nu -2},2^{2\nu -2})$$\end{document} and 2-rank 2ν+2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\nu +2$$\end{document} when ν≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu \ge 3$$\end{document}. For the symplectic graph on 63 vertices we investigate repeated switching by computer and find many new strongly regular graphs with the above parameters for ν=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu =3$$\end{document} with various 2-ranks. Using these results and a recursive construction method for the symplectic graph from Hadamard matrices, we obtain several graphs with the above parameters, but different 2-ranks for every ν≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu \ge 3$$\end{document}.

non-isomorphic with the original graph. In this paper we use the 2-rank of the adjacency matrix to prove non-isomorphism after switching.
When ν = 1 we have the complete graph K 3 , and Sp(4, 2) is a strongly regular graph with parameters (15,8,4,4), which is known to be determined by its parameters. For ν ≥ 3 we find Godsil-McKay switching sets in Sp(2ν, 2) and we prove that the 2-rank increases after switching, which implies that the switched graph is non-isomorphic with the original graph.
It turns out that for ν ≥ 3 the symplectic graph has many switching sets that remain switching sets after switching. Therefore it is interesting to find out what happens after several switchings. We investigated this by computer for Sp (6,2) and found 1826 new strongly regular graphs with parameters (63, 32, 16, 16). For the 2-rank of these new graphs we found six different values.
The symplectic graphs will be defined below, and in Sect. 4 we give an alternative description by use of a well-known recursive construction of Hadamard matrices. We settle the behavior of the 2-ranks of this recursive construction, and apply it to the strongly regular graphs with various 2-ranks found by computer. As a result we find that for every ν ≥ 3 there exist strongly regular graphs with the same parameters as Sp(2ν, 2) for a number of distinct values for the 2-rank. Moreover, this number of different 2-ranks is nondecreasing and goes to infinity when ν → ∞.

The symplectic graphs over F 2
Let F 2ν 2 be the 2ν-dimensional vector space over F 2 , and let K = I ν ⊗ (J 2 − I 2 ), where I ν is the identity matrix of order ν, and J denotes the all-ones matrix of order 2. The symplectic graph Sp(2ν, 2) over F 2 is the graph whose vertices are the nonzero vectors of F 2ν 2 , where two vertices x and y are adjacent whenever x K y = 1. Equivalently, For ν ≥ 2, it is known (see for example [7]) that the symplectic graph Sp(2ν, 2) is a strongly regular graph with parameters 2 2ν − 1, 2 2ν−1 , 2 2ν−2 , 2 2ν−2 .

Godsil-McKay switching and its 2-rank behavior
Godsil and McKay [4] proved the following result. Theorem 1.2 Let G be a graph and let S be a subset of the vertex set of G which induces a regular subgraph. Assume that each vertex outside S is adjacent to |S|, 1 2 |S| or 0 vertices of S. Make a new graph G from G as follows. For each vertex v outside S with 1 2 |S| neighbors in S, delete the 1 2 |S| edges between v and S, and join v instead to the 1 2 |S| other vertices in S. Then G and G have the same spectrum.
The operation that changes G into G is called Godsil-McKay switching. The subset S of the vertex set of G will be called a (Godsil-McKay) switching set. Note that any vertex subset of G of size 2 satisfies the required conditions, but in this case the switched graph G is isomorphic with G. Therefore we assume that a switching set has at least four vertices.
Let A and A be the adjacency matrices of G and G , respectively, and assume that the first |S| rows (and columns) of A and A correspond to the switching set S and the last h rows correspond to the vertices outside S with exactly 1 2 |S| neighbors in S. Then and J is the |S| × h all-ones matrix. Since 2-rank(K ) = 2, the 2-ranks of A and A differ by at most 2. It is well-known that the 2-rank of any adjacency matrix is even (see [3]), thus we have the following result.

Switched symplectic graphs
For ν ≥ 3, we define the following vectors in F 2ν 2 : where z is an arbitrary vector in F 2ν−6 2 .

Proposition 2.1 The set S
Proof Any two vertices from S are nonadjacent, so the subgraph of Sp(2ν, 2) induced by S is a coclique, and therefore regular. Consider an arbitrary vertex x / ∈ S. Then This implies that the number of edges between x and S is even, and therefore S is a switching set.
Let G be the graph obtained from G = Sp(2ν, 2) by switching with respect to S. We shall now prove that G and G are non-isomorphic.
Theorem 2.2 For ν ≥ 3, the graph G obtained from Sp(2ν, 2) by switching with respect to the switching set S given above, is strongly regular with the same parameters as Sp(2ν, 2), but with 2-rank equal to 2ν + 2.
Proof Let A be the adjacency matrix of G = Sp(2ν, 2), and assume that the first four rows and columns correspond to S. Then 2-rank(A) = 2ν and A has 2 2ν − 1 rows. This implies that, over F 2 , every possible nonzero linear combination of a basis of the row space of A is a row of A. Therefore the sum (mod 2) of any two rows of A is again a row of A. Let r 5 and r 6 be rows of A corresponding to the vertices v 5 = [100000z ] and v 6 = [001000z ] , respectively. Then r 5 starts with 0011 and r 6 starts with 0101. It follows that r 7 = r 5 + r 6 is also a row of A starting with 0110. After switching only the first four entries of r 5 , r 6 and r 7 change and become 1100, 1010 and 1001, respectively. Let r i denote the switched version of r i (i = 5, 6 or 7). Then r 5 +r 6 +r 7 = 11110 . . . 0, and this is in the row space of the switched matrix A , but it is not a row of A . So G is not isomorphic to G, and by Theorem 1.1 and Proposition 1.3 the 2-rank of A equals 2ν + 2.
The switching set S given above, is not the only one. There are many more and many remain a switching set after switching with respect to S. (Indeed, for any three independent vectors v 1 , v 2 and v 3 ∈ F 2ν 2 , the set {v 1 , v 2 , v 3 , v 1 + v 2 + v 3 } is a Godsil-McKay switching set whenever the four vertices induce a regular subgraph.) Therefore we can apply switching several times. However it is not true in general that a second switching increases the 2-rank again, and it looks difficult to make a general statement like in the above theorem. Instead we investigated the repeated switching by computer for the case ν = 3.

Repeated switching in Sp(6, 2)
In this section we show that Godsil-McKay switching generates a significant number of nonisomorphic graphs with the same parameters as the symplectic graph Sp(6, 2). By computer we search for all switching sets of size 4 in Sp (6,2). We switch and compute the 2-rank. With the firstly encountered graph for which the 2-rank has increased, we repeat the procedure. We stop if the 2-rank cannot be increased. By this procedure we obtained 1827 non-isomorphic graphs with the parameters of Sp(6, 2). The possible 2-ranks are: 6, 8, 10, 12, 14, 16 and 18. No doubt we would have obtained many more non-isomorphic graphs with these parameters if we would have continued the search for other graphs for which the 2-rank has increased after switching. But the isomorphism tests are very time consuming, and since we are mainly interested in the 2-ranks, we choose not to do so. We did, however, continue with some other graphs without worrying about isomorphism in the hope to find examples with a 2-rank of 20 (or more), without success.
We will not display all newly obtained strongly regular graphs, instead we just give the sequence of switching sets that increases the 2-rank in each step (vertices are represented as row vectors): The number of cospectral graphs obtained in each of the six above iterations is: 4275 with 2-rank 8 (161 are non-isomorphic), 2238 with 2-rank 10 (195 are non-isomorphic), 1242 with 2-rank 12 (301 are non-isomorphic), 818 with 2-rank 14 (489 are non-isomorphic), 508 with 2-rank 16 (508 are non-isomorphic) and 172 with 2-rank 18 (172 are non-isomorphic).
Thus we see that there is still a gap between the constructed cases and the theoretic upper bound for the 2-rank mentioned in Sect. 2, which for Sp (6,2) is 26.

Hadamard matrices and 2-ranks
We recall some results of Hadamard matrices. A square (+1, −1)-matrix H of order n is a Hadamard matrix (or H -matrix) whenever H H = n I . For example is a Hadamard matrix of order 4 (we write − instead of −1). If a row or a column of a Hadamard matrix is multiplied by −1, it remains a Hadamard matrix. We can multiply rows and columns of any Hadamard matrix by −1 such that the first row and column consist of all ones. Such a Hadamard matrix is called normalized. A Hadamard matrix H is said to be graphical if H is symmetric and it has constant diagonal. Note that if H is a graphical Hadamard matrix of order n with δ on the diagonal, then A = 1 2 (J − δ H ) is the adjacency matrix of a graph on n vertices. If H is normalized, the obtained graph has an isolated vertex, and it is well-known that for n > 4 the graph on the remaining n − 1 vertices is strongly regular with parameters (n − 1, n/2, n/4, n/4). And conversely, any strongly regular graph with the above parameters comes from a graphical Hadamard matrix. For example, the above Hadamard matrix H is graphical and normalized. The corresponding graph is the smallest symplectic graph Sp(2, 2) = K 3  Proof It is easily seen that For any integer matrix A we have 2-rank(J ⊗ A) = 2-rank(A ⊗ J ) = 2-rank(A). Therefore To prove the second statement, we define V i to be a matrix consisting of ρ(H i ) independent columns of A H i for i = 1, 2 (so the columns of V 1 and V 2 form a basis for the column space of A H 1 and A H 2 , respectively). Suppose H 1 and H 2 are normalized. Then A H 1 ⊗H 2 contains the columns of 1 ⊗ V 1 and V 2 ⊗ 1. These ρ(H 1 ) + ρ(H 2 ) columns are independent (indeed, the first rows of V 1 and V 2 are all-zero rows and therefore the only vector in the intersection of the column space of 1 ⊗ V 1 and the column space of V 2 ⊗ 1 is the zero vector), and hence ρ( With the Hadamard matrix H of order 4 given above we define Then clearly H ⊗ν is a normalized graphical Hadamard matrix of order 4 ν , and Lemma 4.1 implies that 2-rank(A H ⊗ν ) = ρ(H ⊗ν ) = 2ν. Therefore, by Theorem 1.1 the strongly regular graph associated with H ⊗ν is the symplectic graph Sp(2ν, 2).
In the definition of H ⊗ν we can replace any triple product H ⊗ H ⊗ H by any other regular graphical Hadamard matrix of order 64. By choosing Hadamard matrices coming from the strongly regular graphs with various 2-ranks found by computer in Sect. 3, we obtain normalized graphical Hadamard matrices of order 4 ν , and the 2-rank of the associated strongly regular can take all even values between 2ν and 2ν + 12 ν/3 . Thus we find: Theorem 4.2 For any even r ∈ [2ν, 2ν + 12 ν/3 ] there exists a strongly regular graph with parameters (2 2ν − 1, 2 ν−1 , 2 ν−2 , 2 ν−2 ) and 2-rank r .
Another application of Lemma 4.1 is the following. There exist strongly regular graphs with parameters (35, 18, 9, 9) for the 2-ranks 6, 8, 10, 12 and 14; see [5]. Let H * be the associated normalized graphical Hadamard matrix of order 36, and let H be as before, then H ⊗ H * is associated with a strongly regular graph with parameters (143, 72, 36, 36). By Lemma 4.1 we find that such strongly regular graphs exist for every even 2-rank between 8 and 16.

Remarks
A different construction of graphs with the same parameters as Sp(2ν, 2) was given by Munemasa and Vanhove [6]. It would be interesting to know the 2-rank of their construction. It is claimed in [6] that the construction admits a cyclic difference set, and using Corollary 3.7 from [1], it follows that the 2-rank is a multiple of 2ν, and therefore at least 4ν. So we can conclude that their graphs are not isomorphic to the ones obtained in Theorem 2.2.
A graph associated with a normalized graphical Hadamard matrix is a so-called (v, k, λ) graph, which means that the adjacency matrix can be interpreted as the incidence matrix of a symmetric 2-(v, k, λ) design. It is possible that non-isomorphic graphs lead to isomorphic designs. However, if the matrices have different 2-ranks, then obviously also the designs are non-isomorphic. Thus we can conclude by Theorem 4.2 that there exist at least 1 + 6 ν/3 non-isomorphic symmetric 2-(2 2ν − 1, 2 2ν−1 , 2 2ν−2 ) designs with distinguishing 2-ranks.