Minimum supports of eigenfunctions of graphs Alexandr Valyuzhenich - PowerPoint PPT Presentation

Minimum supports of eigenfunctions of graphs Alexandr Valyuzhenich Sobolev Institute of Mathematics, Novosibirsk G2D2, Yichang, China, August 25, 2019 Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

Outline 1 Basic definitions 2 Minimum support problem (MS-problem) 3 MS-problem for Hamming graphs 4 MS-problem for some other distance-regular graphs 5 MS-problem for the Star graph 6 Open problems Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

Basic definitions Let Γ = ( V , E ) be a graph with the adjacency matrix A . The set of neighbours of a vertex x ∈ V is denoted by N ( x ) . Let λ be an eigenvalue of the matrix A . Definition A function f : V − → R is called a λ -eigenfunction of Γ if f �≡ 0 and the equality � λ · f ( x ) = f ( y ) y ∈ N ( x ) holds for any x ∈ V . Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

Example Figure: 1-eigenfunction of the Petersen graph. Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

Basic definitions Let Γ = ( V , E ) be a graph. Definition The support of a function f : V − → R is the set Supp ( f ) = { x ∈ V | f ( x ) � = 0 } . Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

MS-problem MS-problem For a graph Γ and its eigenvalue λ to find the minimum cardinality of the support of a λ -eigenfunction of Γ . A λ -eigenfunction f of a graph Γ is called extremal if f has the minimum cardinality of the support among all λ -eigenfunctions of Γ . Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

MS-problem In many cases MS-problem is directly related to the problem of finding the minimum possible difference of two combinatorial objects and to the problem of finding the minimum cardinality of the bitrades. H. L. Hwang, On the structure of ( v , k , t ) trades, Journal of Statistical Planning and Inference, 13 (1986) 179–191. T. Etzion, A. Vardy, Perfect binary codes: Constructions, properties and enumeration, IEEE Trans. Inf. Theory, 40(3) (1994) 754–763 D. S. Krotov, I. Yu. Mogilnykh, V. N. Potapov, To the theory of q-ary Steiner and other-type trades, Discrete Mathematics 339(3) (2016) 1150–1157. Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

MS-problem MS-problem has been studied for the following families of graphs: bilinear forms graphs (Sotnikova, 2019) cubical distance-regular graphs (Sotnikova, 2018) Doob graphs (Bespalov, 2018) Grassmann graphs (Krotov, Mogilnykh, Potapov, 2016) Hamming graphs (Potapov 2012; Vorob’ev, Krotov, 2015; Krotov 2016; Valyuzhenich 2017; Valyuzhenich, Vorobev, 2019; Valyuzhenich 2019+) Johnson graphs (Vorob’ev, Mogilnykh, Valyuzhenich, 2018) Paley graphs (Goryainov, Kabanov, Shalaginov, Valyuzhenich, 2018) Star graphs (Goryainov, Kabanov, Konstantinova, Shalaginov, Valyuzhenich, 2019+) Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

Hamming graph Σ q = { 0 , 1 , . . . , q − 1 } . Definition The Hamming graph H ( n , q ) is defined as follows: the vertices of H ( n , q ) are Σ n q . two vertices are adjacent if they differ in exactly one position Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

Hamming graph The eigenvalues of H ( n , q ) are { λ i ( n , q ) = n ( q − 1 ) − q · i | i = 0 , 1 , . . . , n } . Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

MS-problem for the Hamming graph. Case i ≤ n 2 . Theorem 1 ([1, Corollary 2]) For i ≤ n 2 the minimum cardinality of the support of a λ i ( n , 2 ) -eigenfunction of H ( n , 2 ) is 2 n − i . Theorem 2 ([2, Theorem 1]) For q ≥ 3 and i ≤ n 2 the minimum cardinality of the support of a λ i ( n , q ) -eigenfunction of H ( n , q ) is 2 i · ( q − 1 ) i · q n − 2 i . Combining Theorem 1 and Theorem 2, we obtain that for q ≥ 2 and i ≤ n 2 the minimum cardinality of the support of a λ i ( n , q ) -eigenfunction of H ( n , q ) is 2 i · ( q − 1 ) i · q n − 2 i . [1] V. N. Potapov, On perfect 2-colorings of the q-ary n-cube, Discrete Mathematics 312(6) (2012) 1269–1272. [2] A. Valyuzhenich, K. Vorob’ev, Minimum supports of functions on the Hamming graphs with spectral constraints, Discrete Mathematics 342(5) (2019) 1351–1360. Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

MS-problem for the Hamming graph. Case i > n 2 . Theorem 3 ([1, Corollary 2]) For i > n 2 the minimum cardinality of the support of a λ i ( n , 2 ) -eigenfunction of H ( n , 2 ) is 2 i . Theorem 4 ([2, Theorem 3]) For q ≥ 4 and i > n 2 the minimum cardinality of the support of a λ i ( n , q ) -eigenfunction of H ( n , q ) is 2 i · ( q − 1 ) n − i . Combining Theorem 3 and Theorem 4, we obtain that for q ≥ 2 ( q � = 3) and i > n 2 the minimum cardinality of the support of a λ i ( n , q ) -eigenfunction of H ( n , q ) is 2 i · ( q − 1 ) n − i . [1] V. N. Potapov, On perfect 2-colorings of the q-ary n-cube, Discrete Mathematics 312(6) (2012) 1269–1272. [2] A. Valyuzhenich, K. Vorob’ev, Minimum supports of functions on the Hamming graphs with spectral constraints, Discrete Mathematics 342(5) (2019) 1351–1360. Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

Characterization of extremal λ i ( n , q ) -eigenfunctions of H ( n , q ) . So, MS-problem for the Hamming graph is solved for q ≥ 2 and i ≤ n 2 and for q ≥ 2 ( q � = 3) and i > n 2 . Moreover, there was obtained a characterization of extremal λ i ( n , q ) -eigenfunctions of H ( n , q ) . It was proved that such an eigenfunction can be represented as the tensor product of certain elementary extremal eigenfunctions of the Hamming graphs of dimensions not greater that 2. Let us introduce those elementary extremal eigenfunctions. Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

Elementary extremal eigenfunctions in H ( 2 , q ) For all k , m ∈ Σ q , we define a function a k , m , q : Σ 2 q − → R as follows:  1 , if x = k and y � = m ;   a k , m , q ( x , y ) = − 1 , if y = m and x � = k ;  0 , otherwise.  The set of functions a k , m , q , where k , m ∈ Σ q , is denoted by A q . Example. Here are the functions a 1 , 1 , 3 and a 2 , 2 , 3 : We note that any function from A q is a ( q − 2 ) -eigenfunction of H ( 2 , q ) and the cardinality of its support is 2 ( q − 1 ) . Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

Elementary extremal eigenfunctions in H ( 1 , q ) Let B q = { e q } , where e q : Σ q − → R and e q ≡ 1. For all k , m ∈ Σ q and k � = m we define a function c k , m , q : Σ q − → R by the rule:  1 , if x = k ;   c k , m , q ( x ) = − 1 , if x = m ;  0 , otherwise.  The set of functions c k , m , q , where k , m ∈ Σ q and k � = m , is denoted by C q . Example. Here are the functions c 2 , 0 , 3 and c 2 , 1 , 3 : Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

Basic definitions The Cartesian product G � H of graphs G and H is a graph with the vertex set V ( G ) × V ( H ) ; and any two vertices ( u , u ′ ) and ( v , v ′ ) are adjacent if and only if either u = v and u ′ is adjacent to v ′ in H , or u ′ = v ′ and u is adjacent to v in G . Let G = G 1 � G 2 , f 1 : V ( G 1 ) − → R and f 2 : V ( G 2 ) − → R . We define the tensor product f 1 · f 2 by the following rule: ( f 1 · f 2 )( x , y ) = f 1 ( x ) f 2 ( y ) for ( x , y ) ∈ V ( G ) . Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

Extremal λ i ( n , q ) -eigenfunctions of H ( n , q ) for i ≤ n 2 An arbitrary extremal λ i ( n , q ) -eigenfunction of H ( n , q ) , where q ≥ 2 and i ≤ n 2 , is the tensor product of i functions from A q and n − 2 i functions from B q . Example. A typical extremal λ 2 ( 5 , 3 ) -eigenfunction of H ( 5 , 3 ) is the tensor product of the following functions: Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

Extremal λ i ( n , q ) -eigenfunctions of H ( n , q ) for i > n 2 An arbitrary extremal λ i ( n , q ) -eigenfunction of H ( n , q ) , where q ≥ 5 or q = 2 and i > n 2 , is the tensor product of n − i functions from A q and 2 i − n functions from C q . Example. A typical extremal λ 3 ( 4 , 5 ) -eigenfunction of H ( 4 , 5 ) is the tensor product of the following functions: Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

MS-problem for the Hamming graph. Case q = 3 and i > n 2 . Theorem 5 (V., 2019+) 2 < i ≤ 2 n For n 3 the minimum cardinality of the support of a λ i ( n , 3 ) -eigenfunction of H ( n , 3 ) is 2 3 ( n − i ) − i · 3 2 i − n . Theorem 6 (V., 2019+) For i > 2 n 3 the minimum cardinality of the support of a λ i ( n , 3 ) -eigenfunction of H ( n , 3 ) is 2 2 i − n · 3 n − i . In this case we do not have a characterization of extremal eigenfunctions, but we can construct some examples of extremal eigenfunctions. Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

Extremal eigenfunctions in H ( 3 , 3 ) The function on the picture is denoted by h . We note that h is an extremal λ 2 ( 3 , 3 ) -eigenfunction (0-eigenfunction) of H ( 3 , 3 ) and | Supp ( h ) | = 6. Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs