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 Introduction 1 Definitions Bounds Bounded smallest eigenvalue 2 Hoffman’s result The new bound 3 The maximal clique polynomial Some other recent results 4 Sesqui-regular graphs The Lemmens-Seidel Conjecture
Introduction Bounded smallest eigenvalue The new bound Some other recent results Outline Introduction 1 Definitions Bounds Bounded smallest eigenvalue 2 Hoffman’s result The new bound 3 The maximal clique polynomial Some other recent results 4 Sesqui-regular graphs The Lemmens-Seidel Conjecture
Introduction Bounded smallest eigenvalue The new bound Some other recent results Graph � V � G = ( V , E ) with vertex set V and edge set E ⊆ . 2 Adjacency matrix A is a V × V matrix: A x , 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 � V � G = ( V , E ) with vertex set V and edge set E ⊆ . 2 Adjacency matrix A is a V × V matrix: A x , 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 λ min .
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 Introduction 1 Definitions Bounds Bounded smallest eigenvalue 2 Hoffman’s result The new bound 3 The maximal clique polynomial Some other recent results 4 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 . c bound: α ( G ) ≤ min( n − n + , n − n − ) . Cvetkovi´
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 . c bound: α ( G ) ≤ min( n − n + , n − n − ) . Cvetkovi´ 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 . c bound: α ( G ) ≤ min( n − n + , n − n − ) . Cvetkovi´ 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 . c bound: α ( G ) ≤ min( n − n + , n − n − ) . Cvetkovi´ 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 . c bound: α ( G ) ≤ min( n − n + , n − n − ) . Cvetkovi´ 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 . c bound: α ( G ) ≤ min( n − n + , n − n − ) . Cvetkovi´ 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 . c bound: α ( G ) ≤ min( n − n + , n − n − ) . Cvetkovi´ 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 2 k -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 λ min 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 λ min be the smallest eigenvalue of G . n Hoffman bound: α ( G ) ≤ . k 1 + − λ min
Introduction Bounded smallest eigenvalue The new bound Some other recent results Hoffman bound Let G be a k -regular graph of order n . Let λ min be the smallest eigenvalue of G . n Hoffman bound: α ( G ) ≤ . k 1 + − λ min 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 on 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 on 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 k eigenvalue λ min , then ω ( G ) ≤ 1 + − λ min .
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