Computer Algebra Investigation of Known Primitive Triangle-Free Strongly Regular Graphs

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1 Computer Algebra Investigation of Known Primitive Triangle-Free Strongly Regular Graphs Matan Ziv-Av (Jointly with Mikhail Klin) Ben-Gurion University of the Negev SCSS 2013 RISC, JKU July 5, 2013 Ziv-Av (BGU) Triangle free SRGs SCSS / 39

2 1 Preliminaries 2 Triangle free strongly regular graphs 3 tfsrgs inside tfsrgs 4 Equitable partitions of tfsrgs 5 Open questions Ziv-Av (BGU) Triangle free SRGs SCSS / 39

3 1 Preliminaries 2 Triangle free strongly regular graphs 3 tfsrgs inside tfsrgs 4 Equitable partitions of tfsrgs 5 Open questions Ziv-Av (BGU) Triangle free SRGs SCSS / 39

4 Graphs In this talk, a graph is a pair (V, E) of sets. The (finite) set V is called the set of vertices. The set E is called the set of edges. Its elements are subsets of V of size 2. In other words we are discussing simple graphs - undirected, with no loops and no multiple edges. The order of the graph is V, number of vertices. Two vertices are called adjacent or neighbors if they belong to the same edge. The valency of a vertex is the number of its neighbors. A graph is regular if all vertices have the same valency. Ziv-Av (BGU) Triangle free SRGs SCSS / 39

5 Adjacency matrices For a graph Γ = (V, E) of order n, we define adjacency matrix A(Γ). A is a 0, 1-matrix of dimensions n n. The rows and columns are denoted by vertices of Γ in the same order, and the i, j element is 1 if i and j are adjacent and 0 otherwise. All elements of the diagonal are zero. A(Γ) is symmetric. Its spectrum is, therefore, real. Ziv-Av (BGU) Triangle free SRGs SCSS / 39

6 Automorphism groups A permutation π Sym(V ) is called an automorphism of a graph Γ = (V, E) if it maps edges to edges and non-edges to non-edges. In other words π Sym(V ) is an automorphism of Γ if {i, j} E if an only if {i π, j π } E for all i, j V. In matrix language π is an automorphism of Γ if its permutation matrix M π commutes with A(Γ). Aut(Γ), the set of all automorphisms of Γ is a subgroup of Sym(V ). Ziv-Av (BGU) Triangle free SRGs SCSS / 39

7 Strongly regular graphs A graph Γ is called strongly regular graph (SRG) with parameters (v, k, λ, µ) if: Γ is a regular graph of valency k; any two adjacent vertices have λ common neighbors; and any two non-adjacent vertices have µ common neighbors. In matrix language: If A = A(Γ) is the adjacency matrix of Γ, then Γ is an SRG if A 2 = ki + λa + µ(j I A) Ziv-Av (BGU) Triangle free SRGs SCSS / 39

8 Strongly regular graphs Example (Square) SRG(4, 2, 0, 2) Example (Pentagon) SRG(5, 2, 0, 1) Ziv-Av (BGU) Triangle free SRGs SCSS / 39

9 Strongly regular graphs Example (Hexagon) Not strongly regular. Ziv-Av (BGU) Triangle free SRGs SCSS / 39

10 Strongly regular graphs Example (Hexagon) Not strongly regular. Some non-neighbors have one common neighbor. Ziv-Av (BGU) Triangle free SRGs SCSS / 39

11 Strongly regular graphs Example (Hexagon) Not strongly regular. Other non-neighbors have none. Ziv-Av (BGU) Triangle free SRGs SCSS / 39

12 Equitable partitions An equitable partition (EP) of a graph Γ = (V, E) is a partition P 1, P 2,..., P r of its vertex set V such that for every two sets P i, P j, the number of neighbors that a vertex of P i has in P j does not depend on the selection of vertex of P i. The sets of a partition are usually called cells. The equitable closure of a partition P is the equitable partition Q such that every equitable partition that is finer than P is also finer than Q. The finest equitable partition has each vertex in its own cell. For a regular graph, the coarsest equitable partition has one cell. The orbits of a group of automorphisms form an equitable partition of a graph. Such equitable partition is called automorphic. Ziv-Av (BGU) Triangle free SRGs SCSS / 39

13 Collapsed adjacency matrix The quotient (or collapsed) adjacency matrix of an equitable partition is defined as the r r matrix B = (b ij ), where b ij is the number of neighbors that a vertex of P i has in P j. If B is a collapsed adjacency matrix then P i b ij = P j b ji. The characteristic (minimal) polynomial of the quotient adjacency matrix of an equitable partition divides the characteristic (minimal) polynomial of the adjacency matrix of the graph. We may also represent the quotient matrix as an intersection diagram. Ziv-Av (BGU) Triangle free SRGs SCSS / 39

14 Example (Hexagon) An equitable partition {{1, 2, 4, 5}, {3, 6}}. ( ) 1 1 Quotient matrix: Intersection diagram: This EP is automorphic: H = (1, 4)(2, 5)(3, 6), (2, 4)(1, 5). Ziv-Av (BGU) Triangle free SRGs SCSS / 39

15 1 Preliminaries 2 Triangle free strongly regular graphs 3 tfsrgs inside tfsrgs 4 Equitable partitions of tfsrgs 5 Open questions Ziv-Av (BGU) Triangle free SRGs SCSS / 39

16 Triangle free strongly regular graphs An SRG without triangles is called triangle free strongly regular graph (tfsrg). This is the same as requiring λ = 0. An SRG is called imprimitive if it, or its complement is disconnected. An SRG is primitive if µ = 0 or µ = k 1. We are interested in primitive SRGs. In other words, 1 µ k 2. Ziv-Av (BGU) Triangle free SRGs SCSS / 39

17 Moore graphs, µ = 1 An extremal case of tfsrgs is µ = 1. That is, SRGs with no triangles or quadrangles. These graphs are called Moore graphs. Possible only for k = 2, 3, 7, 57. v = k There exists unique graph for k = 2, 3, 7. Existence of Moore graph with k = 57 is one of main open questions in algebraic graph theory. Ziv-Av (BGU) Triangle free SRGs SCSS / 39

18 The known primitive tfsrgs There are seven known primitive triangle free strongly regular graphs. All of them have no more than 100 vertices. All seven known graphs are subgraphs of the largest one, NL 2 (10) on 100 vertices. v k λ µ Name pentagon Petersen Clebsch (NL 1 (4)) Hoffman-Singleton Gewirtz Mesner Higman-Sims (NL 2 (10)) Ziv-Av (BGU) Triangle free SRGs SCSS / 39

19 (5, 2, 0, 1) Pentagon Aut(Pentagon) = D 5 of order 10. Ziv-Av (BGU) Triangle free SRGs SCSS / 39

20 (10, 3, 0, 1) Petersen graph Aut(Petersen) = S 5 of order 120. Pentagon is a subgraph of Petersen graph. Ziv-Av (BGU) Triangle free SRGs SCSS / 39

21 (16, 5, 0, 2) Clebsch graph Clebsch graph 5 - Cayley graph Cay(E 16, {0001, 0010, 0100, 1000, 1111}) (the 4 dimensional cube Q 4 plus diagonals). It is NL 1 (4). Aut( 5 ) = E 2 4 S 5 of order Petersen graph is the induced subgraph on non-neighbors of a vertex. Ziv-Av (BGU) Triangle free SRGs SCSS / 39

22 (50, 7, 0, 1) Hoffman-Singleton graph Automorphism group of order Famous model by Neil Robertson. Construction: Five Pentagons and 5 Pentagrams. Vertex i of Pentagon j is adjacent to vertex hj + i of Pentagram h. Ziv-Av (BGU) Triangle free SRGs SCSS / 39

23 (50, 7, 0, 1) Hoffman-Singleton graph The Robertson model provides two equitable partitions: one into ten pentagons, and the other into 5 ( vertical ) Petersen graphs Ziv-Av (BGU) Triangle free SRGs SCSS / 39

24 (56, 10, 0, 2) Sims-Gewirtz graph From now, there is no sense in showing full diagrams of the graphs. But we present an intersection diagram of an equitable partition. Automorphism group of order 86040, with structure 2 2.L 3 (4). Ziv-Av (BGU) Triangle free SRGs SCSS / 39

25 (77, 16, 0, 4) Mesner graph Automorphism group of order Ziv-Av (BGU) Triangle free SRGs SCSS / 39

26 (100, 22, 0, 6) NL 2 (10), Higman-Sims graph Automorphism group of order Higman-Sims sporadic simple group is a subgroup of index 2 in this group. Ziv-Av (BGU) Triangle free SRGs SCSS / 39

27 1 Preliminaries 2 Triangle free strongly regular graphs 3 tfsrgs inside tfsrgs 4 Equitable partitions of tfsrgs 5 Open questions Ziv-Av (BGU) Triangle free SRGs SCSS / 39

28 Summary We are looking for smaller tfsrgs as subgraphs of NL 2 (10). In case of tfsrgs, a subgraph is an induced subgraph. Easy to see more generally: A subgraph of diameter two in a graph with no triangles is an induced subgraph. All 6 smaller tfsrgs are subgraphs of NL 2 (10). Except for the Petersen graph, all embeddings are unique up to Aut(NL 2 (10)). The picture of such embeddings is, in our eyes, of an independent interest. Ziv-Av (BGU) Triangle free SRGs SCSS / 39

29 Computer results Pentagon Petersen Clebsch HoSi Gewirtz Mesner NL 2 (10) Pentagon Petersen Clebsch HoSi Gewirtz Mesner 1 1 NL 2 (10) 1 Ziv-Av (BGU) Triangle free SRGs SCSS / 39

30 Some interpretation of the results Subgraphs isomorphic to the pentagon can be counted by brute force: construct all sets of 5 vertices and check if induced subgraph is a pentagon. We can also calculate theoretically the number of subgraphs from the parameters of the tfsrg: vk(k 1)(k µ)µ 10. For enumerating all subgraphs isomorphic to Petersen or Clebsch graphs, we used knowledge of pentagons. For example, for finding Petersen graphs, we start with one pentagon P (since all are in the same orbit), and for all other pentagons which are disjoint from P, we check if together with P they induce Petersen graph. For the larger graphs we used the metric partition of the subgraph, together with knowledge of the automorphism group, which leaves in all cases just one possible embedding. Ziv-Av (BGU) Triangle free SRGs SCSS / 39

31 1 Preliminaries 2 Triangle free strongly regular graphs 3 tfsrgs inside tfsrgs 4 Equitable partitions of tfsrgs 5 Open questions Ziv-Av (BGU) Triangle free SRGs SCSS / 39

32 Equitable partitions of small tfsrgs We wish to describe all equitable partitions of Γ (up to Aut(Γ)). Motivation 1: Find patterns to be repeated in putative new graphs. Motivation 2: New models of known graphs. For the Pentagon and Petersen graphs, it is easy to find all by hand. For the Clebsch graph, a brute force search by a computer is feasible. There are about partitions of a set of size 16. But we are only interested in subsets that induce regular subgraphs. This reduces the number of subsets from 2 16 to 1052, and the number of equitable partitions to Ziv-Av (BGU) Triangle free SRGs SCSS / 39

33 Equitable partitions of large tfsrgs For the lager graph the problem is harder. For the Hoffman-Singleton graph we constructed all EPs by using theoretical ideas to limit the search space. For the Gewirtz graph we constructed all non-rigid equitable partitions. For the Mesner graph and NL 2 (10) we constructed all automorphic equitable partitions. Ziv-Av (BGU) Triangle free SRGs SCSS / 39

34 Summary of results Pentagon Petersen Clebsch HoSi Gewirtz Mesner NL 2 (10) EP AEP NR As can be seen in the table, for the four smaller graphs we know all equitable partitions. For all those graphs, the only rigid equitable partition is the trivial one (with all cells of size 1). For the other three graphs, we do not have an example of another rigid equitable partition. Ziv-Av (BGU) Triangle free SRGs SCSS / 39

35 STABGRAPH An algorithm to calculate the equitable closure of a partition. The input is a graph Γ = (V, E) and an ordered partition P = (V 1,..., V r ). 1 For every vertex v V, v is in V k, for every 1 i r, t i = Γ(v) V i (t i is the number of neighbors v has in V i ). define a vector of integers O v = (k, t 1,..., t r ). 2 Sort the set {O v v V } lexicographically. 3 Define a new partition P = (V 1,..., V s ) such that v V j if O v is j th in the ordered list. 4 If the number of cells in P is the same as in P, stop, output is P. 5 P := P. 6 Go to step 1. A simple modification: Add as input a list of sets that should not be split. The algorithm fails if one of those sets is split. With this modification, we can use STABGRAPH to enumerate all partitions finer than a given partition, if the given partition has no large cells (about 35 as a practical limit). Ziv-Av (BGU) Triangle free SRGs SCSS / 39

36 Equitable partitions of Hoffman-Singleton graph Method: Divide and conquer. We divided the equitable partitions into two types: 1 Partitions with a small number of cells; 2 partitions with a small cell. For the first type (at most 5 cells, each of size at least 10), we constructed possible quotient matrices, and for each matrix, found possible equitable partitions. For the second type (a cell of size at most 9), we can enumerate all possible small cells, and for each of them use STABGRAPH, or in some special cases further theoretical arguments, to enumerate all partitions with such a cell. Ziv-Av (BGU) Triangle free SRGs SCSS / 39

37 1 Preliminaries 2 Triangle free strongly regular graphs 3 tfsrgs inside tfsrgs 4 Equitable partitions of tfsrgs 5 Open questions Ziv-Av (BGU) Triangle free SRGs SCSS / 39

38 Looking for new tfsrgs - case (162, 21, 0, 3) Constructing a new triangle free strongly regular graph is an attractive goal. The smallest open case is for a graph with parameters (162, 21, 0, 3). Unlike the smaller graphs with large automorphism groups, there are known limits on the automorphism group of such a graph, if exists (Makhnev & Nosov, Mačaj & Širáň). A computer can help, but a complete brute force search is not feasible. Ziv-Av (BGU) Triangle free SRGs SCSS / 39

39 Looking for new tfsrgs - negative Latin square graphs Mesner described a series of parameters of possible tfsrgs, called negative Latin square graphs, denoted by NL g (g 2 + 3g). NL 1 (4) on 16 = 4 2 vertices is Clebsch graph. NL 2 (10) on 100 = 10 2 vertices is commonly known as Higman-Sims graph. NL 3 (18) does not exist: Kaski and Ostergard showed by computer classification of biplanes on 56 points. Gavrilyuk and Makhnev proved theoretically. NL 4 (28) with parameters (784, 116, 0, 20) is the smallest open case in this series. Trying to construct it using similar techniques to construction of NL 2 (10) requires symmetric 2 (96, 20, 4)-designs. There are many such design. Our initial computer aided attempts yielded negative results. Ziv-Av (BGU) Triangle free SRGs SCSS / 39

40 Looking for new tfsrgs - Moore graph of valency 57 This graph has parameters (3250, 57, 0, 1). Unlike smaller Moore graphs, its automorphism group is not rank 3 (Aschbacher), and not even transitive on vertices (G. Higman). In fact it is quite small, with only a very few possible values for its order (Makhnev & Paduchikh, Mačaj & Širáň). Ziv-Av (BGU) Triangle free SRGs SCSS / 39

41 References These references are related to the open problems, and supplement the reference list in the extended abstract. A. L. Gavrilyuk, A. A. Makhnev. Krein graphs without triangles, Dokl. Akad. Nauk 403 (2005), A.A. Makhnev, V.V. Nosov. On automorphisms of strongly regular graphs with λ = 0 and µ = 3, Algebra i Analiz, 21 (5), 2009, A.A. Makhnev, D.V. Paduchikh. On the automorphism group of the Aschbacher graph. (Russian) Dokl. Akad. Nauk 426 (2009), no. 3, ; translation in Dokl. Math. 79 (2009), no. 3, Ziv-Av (BGU) Triangle free SRGs SCSS / 39

The Gewirtz Graph Morgan J. Rodgers Design Theory Fall 2007

The Gewirtz Graph Morgan J. Rodgers Design Theory Fall 2007 The Gewirtz Graph is the unique strongly regular graph having parameters (56, 10, 0, 2). We will call this graph Γ. This graph was actually discovered