A new bound for cliques in strongly regular graphs Jack Koolen - - PowerPoint PPT Presentation

Introduction Bounded smallest eigenvalue The new bound Some other recent results

A new bound for cliques in strongly regular graphs

Jack Koolen∗

∗School of Mathematical Sciences

University of Science and Technology of China (Based on ongoing joint work with Gary Greaves, and Jongyook Park)

G2D2 August 22, 2019 Dedicated to the 100th birthday anniversary of

• Prof. J.J. (Jaap) Seidel (August 19, 1919 – May 8, 2001)

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Outline

1

Introduction Definitions Bounds

2

Bounded smallest eigenvalue Hoffman’s result

3

The new bound The maximal clique polynomial

4

Some other recent results Sesqui-regular graphs The Lemmens-Seidel Conjecture

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Outline

1

Introduction Definitions Bounds

2

Bounded smallest eigenvalue Hoffman’s result

3

The new bound The maximal clique polynomial

4

Some other recent results Sesqui-regular graphs The Lemmens-Seidel Conjecture

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Graph G = (V, E) with vertex set V and edge set E ⊆ V

2

• .

Adjacency matrix A is a V × V matrix: Ax,y = 1 if xy edge, and 0 otherwise. Eigenvalues The eigenvalues of G are the eigenvalues of A.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Graph G = (V, E) with vertex set V and edge set E ⊆ V

2

• .

Adjacency matrix A is a V × V matrix: Ax,y = 1 if xy edge, and 0 otherwise. Eigenvalues The eigenvalues of G are the eigenvalues of A. In this talk, I will mainly be interested in the smallest eigenvalue of G, denoted by λmin.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Edge-regular and sesqui-regular graphs

Definitions Let G be a connected k-regular graph of order n. G is called edge-regular with parameters (n, k, a), if any two adjacent vertices have exactly a common neighbours. G is called sesqui-regular with parameters (n, k, c), if any two vertices at distance 2 have exactly c common neighbours. G is called co-edge-regular with parameters (n, k, c), if any two distinct non-adjacent vertices have exactly c common neighbours. Blackboard

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Amply-regular and strongly regular graphs

Definitions, continued G is called amply-regular with parameters (n, k, a, c), if G is edge-regular with parameters (n, k, a) and sesqui-regular with parameters (n, k, c). G is called strongly regular (SRG) with parameters (n, k, a, c), if G is edge-regular with parameters (n, k, a) and co-edge-regular with parameters (n, k, c). Examples of strongly regular graphs: The Petersen graph, (t × t)-grid, etc. Black board

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Outline

1

Introduction Definitions Bounds

2

Bounded smallest eigenvalue Hoffman’s result

3

The new bound The maximal clique polynomial

4

Some other recent results Sesqui-regular graphs The Lemmens-Seidel Conjecture

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Let G be a graph of order n. Define α(G) as the maximal cardinality of an independent set (co-clique). Define n+ as the number of positive eigenvalues of G. Define n− as the number of negative eigenvalues of G.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Let G be a graph of order n. Define α(G) as the maximal cardinality of an independent set (co-clique). Define n+ as the number of positive eigenvalues of G. Define n− as the number of negative eigenvalues of G. Cvetkovi´ c bound: α(G) ≤ min(n − n+, n − n−).

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Let G be a graph of order n. Define α(G) as the maximal cardinality of an independent set (co-clique). Define n+ as the number of positive eigenvalues of G. Define n− as the number of negative eigenvalues of G. Cvetkovi´ c bound: α(G) ≤ min(n − n+, n − n−). This bound was mentioned by Qing Xiang in his talk.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Let G be a graph of order n. Define α(G) as the maximal cardinality of an independent set (co-clique). Define n+ as the number of positive eigenvalues of G. Define n− as the number of negative eigenvalues of G. Cvetkovi´ c bound: α(G) ≤ min(n − n+, n − n−). This bound was mentioned by Qing Xiang in his talk. If 0 is an eigenvalue of G, then sometimes you can improve this bound by modifying the adjacency matrix.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Let G be a graph of order n. Define α(G) as the maximal cardinality of an independent set (co-clique). Define n+ as the number of positive eigenvalues of G. Define n− as the number of negative eigenvalues of G. Cvetkovi´ c bound: α(G) ≤ min(n − n+, n − n−). This bound was mentioned by Qing Xiang in his talk. If 0 is an eigenvalue of G, then sometimes you can improve this bound by modifying the adjacency matrix. The Petersen graph has equality in this bound. Black board

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Let G be a graph of order n. Define α(G) as the maximal cardinality of an independent set (co-clique). Define n+ as the number of positive eigenvalues of G. Define n− as the number of negative eigenvalues of G. Cvetkovi´ c bound: α(G) ≤ min(n − n+, n − n−). This bound was mentioned by Qing Xiang in his talk. If 0 is an eigenvalue of G, then sometimes you can improve this bound by modifying the adjacency matrix. The Petersen graph has equality in this bound. Black board Question: α(G) ≤ min(n+, n−)?

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Let G be a graph of order n. Define α(G) as the maximal cardinality of an independent set (co-clique). Define n+ as the number of positive eigenvalues of G. Define n− as the number of negative eigenvalues of G. Cvetkovi´ c bound: α(G) ≤ min(n − n+, n − n−). This bound was mentioned by Qing Xiang in his talk. If 0 is an eigenvalue of G, then sometimes you can improve this bound by modifying the adjacency matrix. The Petersen graph has equality in this bound. Black board Question: α(G) ≤ min(n+, n−)? For primitive SRG, this is true.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Let G be a graph of order n. Define α(G) as the maximal cardinality of an independent set (co-clique). Define n+ as the number of positive eigenvalues of G. Define n− as the number of negative eigenvalues of G. Cvetkovi´ c bound: α(G) ≤ min(n − n+, n − n−). This bound was mentioned by Qing Xiang in his talk. If 0 is an eigenvalue of G, then sometimes you can improve this bound by modifying the adjacency matrix. The Petersen graph has equality in this bound. Black board Question: α(G) ≤ min(n+, n−)? For primitive SRG, this is true. But for 2k-cubes, this is not true.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Hoffman bound

Let G be a k-regular graph of order n. Let λmin be the smallest eigenvalue of G.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Hoffman bound

Let G be a k-regular graph of order n. Let λmin be the smallest eigenvalue of G. Hoffman bound: α(G) ≤

n 1+

k −λmin

.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Hoffman bound

Let G be a k-regular graph of order n. Let λmin be the smallest eigenvalue of G. Hoffman bound: α(G) ≤

n 1+

k −λmin

. Haemers generalized the Hoffman bound to general graphs.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Delsarte bound

Let G be a k-regular graph of order n. Define ω(G) as the maximal order of a clique in G.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Delsarte bound

Let G be a k-regular graph of order n. Define ω(G) as the maximal order of a clique in G. The Cvetkovi´ c bound and the Hoffman bound give bounds

• n ω(G), if you apply them to the complement of G.

This was used in the main result in the talk of Ying-Ying Tan.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Delsarte bound

Let G be a k-regular graph of order n. Define ω(G) as the maximal order of a clique in G. The Cvetkovi´ c bound and the Hoffman bound give bounds

• n ω(G), if you apply them to the complement of G.

This was used in the main result in the talk of Ying-Ying Tan. Delsarte bound: If G is strongly regular with smallest eigenvalue λmin, then ω(G) ≤ 1 +

k −λmin .

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Delsarte bound

Let G be a k-regular graph of order n. Define ω(G) as the maximal order of a clique in G. The Cvetkovi´ c bound and the Hoffman bound give bounds

• n ω(G), if you apply them to the complement of G.

This was used in the main result in the talk of Ying-Ying Tan. Delsarte bound: If G is strongly regular with smallest eigenvalue λmin, then ω(G) ≤ 1 +

k −λmin .

The Delsarte bound and the Hoffman bound give the same bound for SRG. Gary Greaves and Leonard Soicher have shown a slight improvement of the Delsarte bound. Rhys Evans talked about a generalization of their method.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Outline

1

Introduction Definitions Bounds

2

Bounded smallest eigenvalue Hoffman’s result

3

The new bound The maximal clique polynomial

4

Some other recent results Sesqui-regular graphs The Lemmens-Seidel Conjecture

Introduction Bounded smallest eigenvalue The new bound Some other recent results

In this talk I will give a new bound on the clique size of a SRG. It sometimes beats both the Cvetkovi´ c bound and the Delsarte bound.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

In this talk I will give a new bound on the clique size of a SRG. It sometimes beats both the Cvetkovi´ c bound and the Delsarte bound. It can be used to show the non-existence of SRG with certain parameters.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

In this talk I will give a new bound on the clique size of a SRG. It sometimes beats both the Cvetkovi´ c bound and the Delsarte bound. It can be used to show the non-existence of SRG with certain parameters. Before I do so, I will give our main tool.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Note that the smallest eigenvalue of the t-star K1,t is equal to − √ t.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Note that the smallest eigenvalue of the t-star K1,t is equal to − √ t. Now look at the graph ˜

• K2n. This graph is the complete

graph K2n with one extra vertex adjacent to exactly n vertices of the K2n. Black board

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Note that the smallest eigenvalue of the t-star K1,t is equal to − √ t. Now look at the graph ˜

• K2n. This graph is the complete

graph K2n with one extra vertex adjacent to exactly n vertices of the K2n. Black board Its smallest eigenvalue goes to −∞ if n goes to ∞. Black board

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Note that the smallest eigenvalue of the t-star K1,t is equal to − √ t. Now look at the graph ˜

• K2n. This graph is the complete

graph K2n with one extra vertex adjacent to exactly n vertices of the K2n. Black board Its smallest eigenvalue goes to −∞ if n goes to ∞. Black board This shows that, if a graph G has smallest eigenvalue λmin ≥ −λ, then there exists a t = t(λ) such that ˜ K2t and K1,t are not induced subgraphs of G.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Hoffman (1973) showed also the other direction: Assume there exists a t such that ˜ K2t and K1,t are not induced subgraphs of G. Then there exists λ = λ(t) (not depending on G) such that λmin(G) ≥ −λ.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Hoffman (1973) showed also the other direction: Assume there exists a t such that ˜ K2t and K1,t are not induced subgraphs of G. Then there exists λ = λ(t) (not depending on G) such that λmin(G) ≥ −λ. This shows that the smallest eigenvalue very much depend

• n the local structure.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Hoffman (1973) showed also the other direction: Assume there exists a t such that ˜ K2t and K1,t are not induced subgraphs of G. Then there exists λ = λ(t) (not depending on G) such that λmin(G) ≥ −λ. This shows that the smallest eigenvalue very much depend

• n the local structure.

For a proof, see Kim, K., Yang (2016).

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Hoffman (1973) showed also the other direction: Assume there exists a t such that ˜ K2t and K1,t are not induced subgraphs of G. Then there exists λ = λ(t) (not depending on G) such that λmin(G) ≥ −λ. This shows that the smallest eigenvalue very much depend

• n the local structure.

For a proof, see Kim, K., Yang (2016). I think this result led Hoffman to the Hoffman’s limit theorem, as was discussed by Akihiro Munemasa.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Signed graphs

A signed graph (G, σ) is a pair where G is a graph and σ : E(G) → {+1, −1} is a signing of the edges.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Signed graphs

A signed graph (G, σ) is a pair where G is a graph and σ : E(G) → {+1, −1} is a signing of the edges. Adjacency matrix: A is a V × V matrix: Ax,y = σ(xy) if xy edge, and 0 otherwise. Two signed graphs (G, σ) and (H, τ) are switching equivalent, if there exists a diagonal matrix D with ±1 on the diagonal and a permutation matrix P, such that DA(G, σ)D = PTA(H, τ)P.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Signed graphs

A signed graph (G, σ) is a pair where G is a graph and σ : E(G) → {+1, −1} is a signing of the edges. Adjacency matrix: A is a V × V matrix: Ax,y = σ(xy) if xy edge, and 0 otherwise. Two signed graphs (G, σ) and (H, τ) are switching equivalent, if there exists a diagonal matrix D with ±1 on the diagonal and a permutation matrix P, such that DA(G, σ)D = PTA(H, τ)P. It is important to generalize the above result of Hoffman to signed graphs.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Outline

1

Introduction Definitions Bounds

2

Bounded smallest eigenvalue Hoffman’s result

3

The new bound The maximal clique polynomial

4

Some other recent results Sesqui-regular graphs The Lemmens-Seidel Conjecture

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Denote by H(a, t) the graph on a + t + 1 vertices consisting

• f a clique Ka+t together with a vertex that is adjacent to

precisely a vertices of the clique.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Denote by H(a, t) the graph on a + t + 1 vertices consisting

• f a clique Ka+t together with a vertex that is adjacent to

precisely a vertices of the clique. ˜ K2n ≃ H(n, n).

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Denote by H(a, t) the graph on a + t + 1 vertices consisting

• f a clique Ka+t together with a vertex that is adjacent to

precisely a vertices of the clique. ˜ K2n ≃ H(n, n). The graph H(a, t) has an obvious equitable partition with quotient matrix Q =   a 1 a − 1 t a t − 1   .

Introduction Bounded smallest eigenvalue The new bound Some other recent results

It is easy to obtain: Lemma Let Γ be a graph having smallest eigenvalue −m that contains H(a, t) as an induced subgraph. Then (a − m(m − 1))(t − (m − 1)2) (m(m − 1))2. (1)

Introduction Bounded smallest eigenvalue The new bound Some other recent results

The maximal clique polynomial

Using H(a, t), we can obtain: Lemma Let G be an SRG with parameters (v, k, a, c) having smallest eigenvalue −m. Suppose G has a maximal clique of size t such that c <

t+m−1+√ (t+m−1)(t−(m−1)(4m−1)) 2

. Then (2(t − 1)(a − t + 2) − (t + (m − 1))(k − t + 1))2 −(k + 1 − t)2(t + (m − 1))(t − (m − 1)(4m − 1)) 0.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

The maximal clique polynomial

Using H(a, t), we can obtain: Lemma Let G be an SRG with parameters (v, k, a, c) having smallest eigenvalue −m. Suppose G has a maximal clique of size t such that c <

t+m−1+√ (t+m−1)(t−(m−1)(4m−1)) 2

. Then (2(t − 1)(a − t + 2) − (t + (m − 1))(k − t + 1))2 −(k + 1 − t)2(t + (m − 1))(t − (m − 1)(4m − 1)) 0. The meaning is that, if we have a large clique, then, sometimes, it is part of a much larger maximal clique. Black board

Introduction Bounded smallest eigenvalue The new bound Some other recent results

The maximal clique polynomial

Using H(a, t), we can obtain: Lemma Let G be an SRG with parameters (v, k, a, c) having smallest eigenvalue −m. Suppose G has a maximal clique of size t such that c <

t+m−1+√ (t+m−1)(t−(m−1)(4m−1)) 2

. Then (2(t − 1)(a − t + 2) − (t + (m − 1))(k − t + 1))2 −(k + 1 − t)2(t + (m − 1))(t − (m − 1)(4m − 1)) 0. The meaning is that, if we have a large clique, then, sometimes, it is part of a much larger maximal clique. Black board We can show similar results for DRG and amply-regular graphs.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Large cliques

Let G be an SRG with parameters (v, k, a, c) having smallest eigenvalue −m.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Large cliques

Let G be an SRG with parameters (v, k, a, c) having smallest eigenvalue −m. We need to find large cliques. How do we do this?

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Large cliques

Let G be an SRG with parameters (v, k, a, c) having smallest eigenvalue −m. We need to find large cliques. How do we do this? Claw bound: If 2(ℓ(a + 1) − k)/(ℓ(ℓ − 1)) > c − 1, then there is no K1,ℓ. (Folklore)

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Large cliques

Let G be an SRG with parameters (v, k, a, c) having smallest eigenvalue −m. We need to find large cliques. How do we do this? Claw bound: If 2(ℓ(a + 1) − k)/(ℓ(ℓ − 1)) > c − 1, then there is no K1,ℓ. (Folklore) This follows easily by the inclusion-exclusion principle.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Large cliques

Let G be an SRG with parameters (v, k, a, c) having smallest eigenvalue −m. We need to find large cliques. How do we do this? Claw bound: If 2(ℓ(a + 1) − k)/(ℓ(ℓ − 1)) > c − 1, then there is no K1,ℓ. (Folklore) This follows easily by the inclusion-exclusion principle. Suppose that G contains a K1,ℓ, but no K1,ℓ+1.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Large cliques

Let G be an SRG with parameters (v, k, a, c) having smallest eigenvalue −m. We need to find large cliques. How do we do this? Claw bound: If 2(ℓ(a + 1) − k)/(ℓ(ℓ − 1)) > c − 1, then there is no K1,ℓ. (Folklore) This follows easily by the inclusion-exclusion principle. Suppose that G contains a K1,ℓ, but no K1,ℓ+1. Then, it is easy to show that G contains a clique with at least a + 2 − (ℓ − 1)(c − 1) vertices. (Bose?)

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Large cliques

Let G be an SRG with parameters (v, k, a, c) having smallest eigenvalue −m. We need to find large cliques. How do we do this? Claw bound: If 2(ℓ(a + 1) − k)/(ℓ(ℓ − 1)) > c − 1, then there is no K1,ℓ. (Folklore) This follows easily by the inclusion-exclusion principle. Suppose that G contains a K1,ℓ, but no K1,ℓ+1. Then, it is easy to show that G contains a clique with at least a + 2 − (ℓ − 1)(c − 1) vertices. (Bose?)

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Some examples of feasible parameters: v k a c λmin DB CB forbid range Kn 12750 671 184 27 −4 168 305 84 129 82 17500 921 254 37 −4 231 307 71 213 112 23276 1330 372 58 −4 333 285 71 337 146 25025 1426 399 62 −4 357 286 74 364 157

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Outline

1

Introduction Definitions Bounds

2

Bounded smallest eigenvalue Hoffman’s result

3

The new bound The maximal clique polynomial

4

Some other recent results Sesqui-regular graphs The Lemmens-Seidel Conjecture

Introduction Bounded smallest eigenvalue The new bound Some other recent results

The following results were obtained by using the non-existence

• f H(a, t) as induced subgraphs.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

The following results were obtained by using the non-existence

• f H(a, t) as induced subgraphs.The following result was shown

with J.Y. Yang:

Introduction Bounded smallest eigenvalue The new bound Some other recent results

The following results were obtained by using the non-existence

• f H(a, t) as induced subgraphs.The following result was shown

with J.Y. Yang: Theorem Let λ ≥ 1 be a real number. Let G be a connected co-edge regular graph with parameters (v, k, c). Then there exists a real number C(λ) (only depending on λ) such that, if G has smallest eigenvalue at least −λ, then c > C(λ) implies that v − k − 1 ≤ (λ−1)2

4

+ 1 holds.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

The following results were obtained by using the non-existence

• f H(a, t) as induced subgraphs.The following result was shown

with J.Y. Yang: Theorem Let λ ≥ 1 be a real number. Let G be a connected co-edge regular graph with parameters (v, k, c). Then there exists a real number C(λ) (only depending on λ) such that, if G has smallest eigenvalue at least −λ, then c > C(λ) implies that v − k − 1 ≤ (λ−1)2

4

+ 1 holds. Our proof only needed sesqui-regularity.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Let ℓ be an integer at least 3 and let λ := 2 √ ℓ − 1.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Let ℓ be an integer at least 3 and let λ := 2 √ ℓ − 1. Take the infinite family of the bipartite ℓ-regular Ramanujan graphs, as constructed by A. Marcus, D.A. Spielman, and

• N. Srivastava (2015).

The graphs in this family are clearly edge-regular with a = 0 and have second largest eigenvalue at most 2 √ ℓ − 1.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Let ℓ be an integer at least 3 and let λ := 2 √ ℓ − 1. Take the infinite family of the bipartite ℓ-regular Ramanujan graphs, as constructed by A. Marcus, D.A. Spielman, and

• N. Srivastava (2015).

The graphs in this family are clearly edge-regular with a = 0 and have second largest eigenvalue at most 2 √ ℓ − 1. Let Γ be a graph in this family with v vertices. Then the complement of Γ is co-edge-regular with parameters (v, v − ℓ − 1, v − 2ℓ) and has smallest eigenvalue at least −1 − 2 √ ℓ − 1.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Let ℓ be an integer at least 3 and let λ := 2 √ ℓ − 1. Take the infinite family of the bipartite ℓ-regular Ramanujan graphs, as constructed by A. Marcus, D.A. Spielman, and

• N. Srivastava (2015).

The graphs in this family are clearly edge-regular with a = 0 and have second largest eigenvalue at most 2 √ ℓ − 1. Let Γ be a graph in this family with v vertices. Then the complement of Γ is co-edge-regular with parameters (v, v − ℓ − 1, v − 2ℓ) and has smallest eigenvalue at least −1 − 2 √ ℓ − 1. This shows that our conclusion is sharp.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

With B. Gebremichel and J.Y. Yang we improved this to

Introduction Bounded smallest eigenvalue The new bound Some other recent results

With B. Gebremichel and J.Y. Yang we improved this to Theorem Let λ ≥ 1 be an integer. Let G be a connected sesqui-regular graph with parameters (v, k, c). Then there exists a real number κ(λ) (only depending on λ) such that, if G has smallest eigenvalue at least −λ, then k > κ(λ) implies that at least one

• f the following holds:

v − k − 1 ≤ (λ−1)2

4

+ 1; c ≤ (λ − 1)λ2. The block graphs of 2-(v, m, 1)-designs are strongly regular with smallest eigenvalue −m and c = m2.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

With B. Gebremichel and J.Y. Yang we improved this to Theorem Let λ ≥ 1 be an integer. Let G be a connected sesqui-regular graph with parameters (v, k, c). Then there exists a real number κ(λ) (only depending on λ) such that, if G has smallest eigenvalue at least −λ, then k > κ(λ) implies that at least one

• f the following holds:

v − k − 1 ≤ (λ−1)2

4

+ 1; c ≤ (λ − 1)λ2. The block graphs of 2-(v, m, 1)-designs are strongly regular with smallest eigenvalue −m and c = m2. This shows that we can not improve much our upper bound in the second item.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

For m = 3, we (B. Gebremichel, K., M. U. Rehman, J.Y. Yang, Q.Q. Yang) can show that the upper bound in the second item is equal to 9.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

For m = 3, we (B. Gebremichel, K., M. U. Rehman, J.Y. Yang, Q.Q. Yang) can show that the upper bound in the second item is equal to 9. And probably we can show that only the block graphs of Steiner triple systems attain this bound.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

For m = 3, we (B. Gebremichel, K., M. U. Rehman, J.Y. Yang, Q.Q. Yang) can show that the upper bound in the second item is equal to 9. And probably we can show that only the block graphs of Steiner triple systems attain this bound. The proof for this is much harder than the general case.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Outline

1

Introduction Definitions Bounds

2

Bounded smallest eigenvalue Hoffman’s result

3

The new bound The maximal clique polynomial

4

Some other recent results Sesqui-regular graphs The Lemmens-Seidel Conjecture

Introduction Bounded smallest eigenvalue The new bound Some other recent results

The Lemmens-Seidel Conjecture

This is based on ongoing joint work with M. Y. Cao, Y.-C. Lin, W.-H. Yu.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

The Lemmens-Seidel Conjecture

This is based on ongoing joint work with M. Y. Cao, Y.-C. Lin, W.-H. Yu. A Seidel matrix of order n, S, is the adjacency matrix of a (Kn, σ) for some signing σ.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

The Lemmens-Seidel Conjecture

This is based on ongoing joint work with M. Y. Cao, Y.-C. Lin, W.-H. Yu. A Seidel matrix of order n, S, is the adjacency matrix of a (Kn, σ) for some signing σ. There is a close relationship between Seidel matrices and systems of equi-angular lines, see the talk of Wei-Hsuan Yu.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

The Lemmens-Seidel Conjecture

This is based on ongoing joint work with M. Y. Cao, Y.-C. Lin, W.-H. Yu. A Seidel matrix of order n, S, is the adjacency matrix of a (Kn, σ) for some signing σ. There is a close relationship between Seidel matrices and systems of equi-angular lines, see the talk of Wei-Hsuan Yu. Let θmin be the smallest eigenvalue of S.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

The Lemmens-Seidel Conjecture

This is based on ongoing joint work with M. Y. Cao, Y.-C. Lin, W.-H. Yu. A Seidel matrix of order n, S, is the adjacency matrix of a (Kn, σ) for some signing σ. There is a close relationship between Seidel matrices and systems of equi-angular lines, see the talk of Wei-Hsuan Yu. Let θmin be the smallest eigenvalue of S. We are interested in the rank r of S − θminI.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

The Lemmens-Seidel Conjecture

This is based on ongoing joint work with M. Y. Cao, Y.-C. Lin, W.-H. Yu. A Seidel matrix of order n, S, is the adjacency matrix of a (Kn, σ) for some signing σ. There is a close relationship between Seidel matrices and systems of equi-angular lines, see the talk of Wei-Hsuan Yu. Let θmin be the smallest eigenvalue of S. We are interested in the rank r of S − θminI. (Neumann 1973): If n > 2r, then θmin is an odd integer.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Let M(r, θ) denote the maximal order of a Seidel matrix with smallest eigenvalue −θ and rank r, as defined above.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Let M(r, θ) denote the maximal order of a Seidel matrix with smallest eigenvalue −θ and rank r, as defined above. Lemmens and Seidel (1973): If r ≥ 16, then M(r, 3) = 2r − 2.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Let M(r, θ) denote the maximal order of a Seidel matrix with smallest eigenvalue −θ and rank r, as defined above. Lemmens and Seidel (1973): If r ≥ 16, then M(r, 3) = 2r − 2. They conjectured: If r ≥ 185, then M(r, 5) = 3

2(r − 1).

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Let M(r, θ) denote the maximal order of a Seidel matrix with smallest eigenvalue −θ and rank r, as defined above. Lemmens and Seidel (1973): If r ≥ 16, then M(r, 3) = 2r − 2. They conjectured: If r ≥ 185, then M(r, 5) = 3

2(r − 1).

They showed it under some extra conditions. Lin and Yu also gave partial results. See Wei-Hsuan Yu’s talk.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Let M(r, θ) denote the maximal order of a Seidel matrix with smallest eigenvalue −θ and rank r, as defined above. Lemmens and Seidel (1973): If r ≥ 16, then M(r, 3) = 2r − 2. They conjectured: If r ≥ 185, then M(r, 5) = 3

2(r − 1).

They showed it under some extra conditions. Lin and Yu also gave partial results. See Wei-Hsuan Yu’s talk. We showed this conjecture last week. Our main tool: The graphs H(a, t).

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Let M(r, θ) denote the maximal order of a Seidel matrix with smallest eigenvalue −θ and rank r, as defined above. Lemmens and Seidel (1973): If r ≥ 16, then M(r, 3) = 2r − 2. They conjectured: If r ≥ 185, then M(r, 5) = 3

2(r − 1).

They showed it under some extra conditions. Lin and Yu also gave partial results. See Wei-Hsuan Yu’s talk. We showed this conjecture last week. Our main tool: The graphs H(a, t). Balla et al. (2016) conjectured that for fixed odd integer θ = 2τ + 1 and r much larger than θ, one has M(r, 2τ + 1) = τ+1

τ r + o(r).

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Let M(r, θ) denote the maximal order of a Seidel matrix with smallest eigenvalue −θ and rank r, as defined above. Lemmens and Seidel (1973): If r ≥ 16, then M(r, 3) = 2r − 2. They conjectured: If r ≥ 185, then M(r, 5) = 3

2(r − 1).

They showed it under some extra conditions. Lin and Yu also gave partial results. See Wei-Hsuan Yu’s talk. We showed this conjecture last week. Our main tool: The graphs H(a, t). Balla et al. (2016) conjectured that for fixed odd integer θ = 2τ + 1 and r much larger than θ, one has M(r, 2τ + 1) = τ+1

τ r + o(r).

This conjecture was proved by a team of MIT very recently, including Zilin Jiang and two undergraduate students.

Introduction Bounded smallest eigenvalue The new bound Some other recent results

Thank you for your attention.