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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) =

• y∈N(x)

f (y) 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

• f 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

• bjects 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
• ther-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

Σq = {0, 1, . . . , q − 1}. Definition The Hamming graph H(n, q) is defined as follows: the vertices of H(n, q) are Σn

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 {λi(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

λi(n, 2)-eigenfunction of H(n, 2) is 2n−i. Theorem 2 ([2, Theorem 1]) For q ≥ 3 and i ≤ n

2 the minimum cardinality of the support of a

λi(n, q)-eigenfunction of H(n, q) is 2i · (q − 1)i · qn−2i. Combining Theorem 1 and Theorem 2, we obtain that for q ≥ 2 and i ≤ n

2 the minimum cardinality of the support of a

λi(n, q)-eigenfunction of H(n, q) is 2i · (q − 1)i · qn−2i.

[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

λi(n, 2)-eigenfunction of H(n, 2) is 2i. Theorem 4 ([2, Theorem 3]) For q ≥ 4 and i > n

2 the minimum cardinality of the support of a

λi(n, q)-eigenfunction of H(n, q) is 2i · (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

λi(n, q)-eigenfunction of H(n, q) is 2i · (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 λi(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 λi(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 ∈ Σq, we define a function ak,m,q : Σ2

q −

→ R as follows: ak,m,q(x, y) =      1, if x = k and y = m; −1, if y = m and x = k; 0,

• therwise.

The set of functions ak,m,q, where k, m ∈ Σq, is denoted by Aq.

• Example. Here are the functions a1,1,3 and a2,2,3:

We note that any function from Aq 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 Bq = {eq}, where eq : Σq − → R and eq ≡ 1. For all k, m ∈ Σq and k = m we define a function ck,m,q : Σq − → R by the rule: ck,m,q(x) =      1, if x = k; −1, if x = m; 0,

• therwise.

The set of functions ck,m,q, where k, m ∈ Σq and k = m, is denoted by Cq.

• Example. Here are the functions c2,0,3 and c2,1,3:

Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

Basic definitions

The Cartesian product GH 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 = G1G2, f1 : V (G1) − → R and f2 : V (G2) − → R. We define the tensor product f1 · f2 by the following rule: (f1 · f2)(x, y) = f1(x)f2(y) for (x, y) ∈ V (G).

Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

Extremal λi(n, q)-eigenfunctions of H(n, q) for i ≤ n

2

An arbitrary extremal λi(n, q)-eigenfunction of H(n, q), where q ≥ 2 and i ≤ n

2, is the tensor product of i functions from Aq

and n − 2i functions from Bq.

• Example. A typical extremal λ2(5, 3)-eigenfunction of H(5, 3) is

the tensor product of the following functions:

Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

Extremal λi(n, q)-eigenfunctions of H(n, q) for i > n

2

An arbitrary extremal λi(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 Aq and 2i − n functions from Cq.

• Example. A typical extremal λ3(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+) For n

2 < i ≤ 2n 3 the minimum cardinality of the support of a

λi(n, 3)-eigenfunction of H(n, 3) is 23(n−i)−i · 32i−n. Theorem 6 (V., 2019+) For i > 2n

3 the minimum cardinality of the support of a

λi(n, 3)-eigenfunction of H(n, 3) is 22i−n · 3n−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 λ2(3, 3)-eigenfunction (0-eigenfunction) of H(3, 3) and |Supp(h)| = 6.

Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

Examples of extremal eigenfunctions of H(n, q) for q = 3 and i > n

2.

The tensor product of 2n − 3i functions from Aq and 2i − n functions h is an extremal λi(n, 3)-eigenfunction of H(n, 3) for

n 2 < i ≤ 2n 3 .

The tensor product of 3i − 2n functions from Cq and n − i functions h is an extremal λi(n, 3)-eigenfunction of H(n, 3) for i > 2n

3 .

Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

MS-problem for the Hamming graph.

Finally, we have the following results on MS-problem for the Hamming graph.

1 For q ≥ 2 and i ≤ n

2 the minimum cardinality of the support of

a λi(n, q)-eigenfunction of H(n, q) is 2i · (q − 1)i · qn−2i.

2 For q ≥ 2 (q = 3) and i > n

2 the minimum cardinality of the

support of a λi(n, q)-eigenfunction of H(n, q) is 2i · (q − 1)n−i.

3 For n

2 < i ≤ 2n 3 the minimum cardinality of the support of a

λi(n, 3)-eigenfunction of H(n, 3) is 23(n−i)−i · 32i−n.

4 For i > 2n

3 the minimum cardinality of the support of a

λi(n, 3)-eigenfunction of H(n, 3) is 22i−n · 3n−i.

Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

MS-problem for the Hamming graph.

So, MS-problem for the Hamming graph is solved for all cases. On the other hand, we have two open problems on a characterization of extremal eigenfunctions of the Hamming graph:

1 Characterize all extremal λi(n, 3)-eigenfunctions of H(n, 3) for

i > n

2.

2 Characterize all extremal λi(n, 4)-eigenfunctions of H(n, 4) for

i > n

2.

Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

1-perfect bitrade

Definition A pair (T0, T1) of disjoint nonempty subsets of vertices of H(n, q) is called a 1-perfect bitrade if every ball of radius 1 (in H(n, q)) either contains one element from T0 and one element from T1 or does not contain elements from T0 ∪ T1. If (T0, T1) is a 1-perfect bitrade, then the following properties holds: T0 and T1 are independent sets in H(n, q) and |T0| = |T1|. the subgraph of H(n, q) induced by T0 ∪ T1 is a perfect matching. Definition The size of a bitrade (T0, T1) is |T0| + |T1|.

Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

Example of 1-perfect bitrade

Figure: 1-perfect bitrade in H(3, 2).

Let T0 and T1 consist of red and blue vertices respectively. Then (T0, T1) is a 1-perfect bitrade of size 4 in H(3, 2).

Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

Connection between 1-perfect bitrades and 1-perfect codes

Proposition Let C0 and C1 be two 1–perfect codes in H(n, q). Then the pair (C0 \ C1, C1 \ C0) is a 1–perfect bitrade in H(n, q). The opposite is not true in general. There are 1-perfect bitrades that can not be presented as (C0 \ C1, C1 \ C0), where C0 and C1 are 1–perfect codes.

Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

Some facts about 1-perfect bitrades

1 If H(n, q) has a 1-perfect bitrade, then n = qm + 1 for some

m ∈ N.

2 For q = pk, where p is a prime, there are several constructions

• f 1-perfect bitrades in H(qm + 1, q).

3 For q = pk the problem of the existence of 1-perfect bitrades

in H(n, q) is open. The fist open case is n = 7 and q = 6.

• K. V. Vorobev, D. S. Krotov, Bounds for the size of a minimal 1-perfect bitrade in a

Hamming graph, Journal of Applied and Industrial Mathematics 9(1) (2015) 141–146.

Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

Problem 2

Problem 2 Given n ≥ 3 and q ≥ 2, to find the minimum size of a 1-perfect bitrade in H(n, q).

Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

Problem 2

For q = 2 Problem 2 is solved. Theorem ([1, Corollary 2]) The minimum size of a 1-perfect bitrade in H(2m + 1, 2), where m ≥ 1, is 2m+1.

[1] V. N. Potapov, On perfect 2-colorings of the q-ary n-cube, Discrete Mathematics 312(6) (2012) 1269–1272.

For q ≥ 3 Problem 2 is open.

Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

Problem 2

For q = 2 Problem 2 is solved. Theorem ([1, Corollary 2]) The minimum size of a 1-perfect bitrade in H(2m + 1, 2), where m ≥ 1, is 2m+1.

[1] V. N. Potapov, On perfect 2-colorings of the q-ary n-cube, Discrete Mathematics 312(6) (2012) 1269–1272.

For q ≥ 3 Problem 2 is open. Now, using Theorem 6, we solve Problem 2 for q = 3.

Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

Problem 2 for q = 3

Firstly, we show that an arbitrary 1-perfect bitrade can be viewed as (−1)-eigenfunction of H(n, q). For a 1-perfect bitrade (T0, T1) we define the function f(T0,T1) : Σn

q −

→ {−1, 0, 1} by the following rule: f(T0,T1)(x) =      1, if x ∈ T0; −1, if x ∈ T1; 0,

• therwise.

Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

Problem 2 for q = 3

Lemma Let (T0, T1) be a 1-perfect bitrade in H(n, q). Then f(T0,T1) is a (−1)-eigenfunction of H(n, q).

Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

Problem 2 for q = 3

Lemma Let (T0, T1) be a 1-perfect bitrade in H(n, q). Then f(T0,T1) is a (−1)-eigenfunction of H(n, q). Let us consider an arbitrary ball B(x) of radius 1 with the center x in H(n, q). By the definition of a 1-perfect bitrade we have two cases. Case 1. Suppose B(x) contains t0 ∈ T0 and t1 ∈ T1. Then f(T0,T1)(y) = 0 for any y ∈ B(x) \ {t0, t1}, f(T0,T1)(t0) = 1 and f(T0,T1)(t1) = −1. Case 2. Suppose B(x) does not contain vertices from T0 and T1. Then f(T0,T1)(y) = 0 for any y ∈ B(x). So, in any case we have that

y∈B(x) f(T0,T1)(y) = 0.

Consequently f(T0,T1)(x) = −

y∈N(x) f(T0,T1)(y), i.e. f(T0,T1) is a

(−1)-eigenfunction of H(n, q).

Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

Problem 2 for q = 3

Theorem (V., 2019) The minimum size of a 1-perfect bitrade in H(3m + 1, 3), where m ≥ 1, is 2m+1 · 3m. Firstly, we prove the bound. Let (T0, T1) be a 1-perfect bitrade in H(3m + 1, 3). Since f(T0,T1) is a (−1)-eigenfunction of H(3m + 1, 3), by Theorem 6 we obtain that |Supp(f(T0,T1))| ≥ 2m+1 · 3m. Then using the equality |T0| + |T1| = |Supp(f(T0,T1))|, we have |T0| + |T1| ≥ 2m+1 · 3m.

Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

Problem 2 for q = 3

So, it remains to prove that the bound is sharp. We use the following result. Theorem ([3, Proposition 2]) Let n = qm + 1 and q = pk, where p is a prime. Then there exist a 1-perfect bitrade in H(n, q) of size 2m+1 · qm(q−2). Using this theorem for q = 3, we have that the bound |T0| + |T1| ≥ 2m+1 · 3m is sharp. So, we solve Problem 2 for q = 3.

[3] K. V. Vorobev, D. S. Krotov, Bounds for the size of a minimal 1-perfect bitrade in a Hamming graph, Journal of Applied and Industrial Mathematics 9(1) (2015) 141–146.

Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

MS-problem for other distance-regular graphs

MS-problem for the bilinear forms graphs of diameter D = 2

• ver a prime field is solved for the smallest eigenvalue (without

characterization of extremal eigenfunctions).

• E. V. Sotnikova, Minimum supports of eigenfunctions in bilinear forms graphs,

Siberian Electronic Mathematical Reports 16 (2019) 501–515.

MS-problem for the cubical distance-regular graphs is solved for 10 of them (with a characterization of extremal eigenfunctions).

• E. V. Sotnikova, Eigenfunctions supports of minimum cardinality in cubical

distance-regular graphs, Siberian Electronic Mathematical Reports 15 (2018) 223–245.

Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

MS-problem for other distance-regular graphs

MS-problem for the Doob graphs is solved for the second largest and the smallest eigenvalues (with a characterization of extremal eigenfunctions in terms of tensor product).

• E. A. Bespalov, On the minimum supports of some eigenfunctions in the Doob

graphs, Siberian Electronic Mathematical Reports 15 (2018) 258–266.

MS-problem for the Grassmann graphs is solved for the smallest eigenvalue.

• 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 for other distance-regular graphs

MS-problem for the Johnson graphs is solved for all eigenvalues (with a characterization of extremal eigenfunctions).

• K. Vorob’ev, I. Mogilnykh, A. Valyuzhenich, Minimum supports of

eigenfunctions of Johnson graphs, Discrete Mathematics 341(8) (2018) 2151–2158.

MS-problem for the Paley graphs of square order is solved for both non-principal eigenvalues (without characterization of extremal eigenfunctions).

• S. Goryainov, V. Kabanov, L. Shalaginov, A. Valyuzhenich, On eigenfunctions

and maximal cliques of Paley graphs of square order, Finite Fields and Their Applications 52 (2018) 361–369.

Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

Star graph

The Star graph Sn = Cay(Symn, S), where n ≥ 3, is the Cayley graph on the symmetric group Symn with the generating set S = {(1 i) | i ∈ {2, . . . , n}}. The spectrum of the Star graph is integral. More precisely, for n ≥ 4 the eigenvalues of Sn are {−(n − 1), −(n − 2), . . . , −1, 0, 1, . . . , (n − 2), (n − 1)} and the eigenvalues of S3 are {−2, −1, 1, 2}. Since the Star graph is bipartite, mul(n − t) = mul(−n + t) for each integer 1 ≤ t ≤ n.

Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

Multiplicities of eigenvalues of the Star graph

Theorem ([5], Theorem) Let n, t ∈ Z, n ≥ 3 and 1 ≤ t ≤ n+1

2 . Then the multiplicity of the

eigenvalue (n − t) of the Star graph Sn is given by the following formula: mul(n − t) = n2(t−1)

(t−1)! + P(n), where P(n) is a polynomial of degree

2t − 3. Moreover, explicit formulas for calculating multiplicities of eigenvalues ±(n − t), where 2 ≤ t ≤ 12, were found. In particular, mul(n − 2) = mul(2 − n) = (n − 1)(n − 2).

[4] S. V. Avgustinovich, E. N. Khomyakova, E. V. Konstantinova, Multiplicities of eigenvalues of the Star graph, Siberian Electronic Mathematical Report, 13 (2016) 1258–1270. [5] E. N. Khomyakova, On the eigenvalues multiplicity function of the Star graph, Siberian Electronic Mathematical Report, 15 (2018) 1416–1425.

Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

MS-problem for the Star graph

Let i ∈ {1, . . . , n} and k, j ∈ {2, . . . , n}, where k = j. We define the function f k,j

i

: Symn − → R by the following rule: f k,j

i

(π) =      1, if π(k) = i; −1, if π(j) = i; 0,

• therwise.

We note that f k,j

i

is an (n − 2)-eigenfunction of Sn and |Supp(f k,j

i

)| = 2(n − 1)!. Denote F = {f k,j

i

| i ∈ {1, . . . , n}, k, j ∈ {2, . . . , n}, k = j}.

Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

MS-problem for Star graph

Theorem (Goryainov, Kabanov, Konstantinova, Shalaginov, V., 2019+) For n ≥ 8 and n = 3 the minimum cardinality of the support of an (n − 2)-eigenfunction of Sn is 2(n − 1)!. Moreover, for n ≥ 8 and n = 3 an (n − 2)-eigenfunction of Sn f is extremal if and only if f = c · ˜ f , where c is a real nonzero constant and ˜ f ∈ F.

Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

Ideas of the proof.

Firstly, we need the following result. Denote F2 = {f 2,j

i

| i ∈ {2, . . . , n}, j ∈ {3, . . . , n}}. Lemma ([6], Lemma 15]) For n ≥ 3, the set F2 forms a basis of the eigenspace of Sn with eigenvalue n − 2.

[6] S. Goryainov, V. V. Kabanov, E. Konstantinova, L. Shalaginov, A. Valyuzhenich. PI-eigenfunctions of the Star graphs arXiv:1802.06611, February 2018.

Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

Ideas of the proof

Let f be an (n − 2)-eigenfunction of Sn. By Lemma 1 there exist the numbers µj

i(f ) ∈ R, where i ∈ {2, . . . , n} and j ∈ {3, . . . , n},

such that f =

• j∈{3,...,n}

i∈{2,...,n}

µj

i(f ) · f 2,j i

. We define the matrix M(f ) = (mi,j(f ))i,j∈{1,...,n} by the following rule: mi,j(f ) =      −µj

i(f ),

if i > 1 and j > 2; n

s=3 µs i (f ),

if i > 1 and j = 2; 0, if i = 1 or j = 1. Lemma 2 Let f be an (n − 2)-eigenfunction of Sn. Then f (π) = n

i=1 mi,π−1(i)(f ) for any π ∈ Symn.

Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

Ideas of the proof

Then E(f ) = {

n

• i=1

mi,π(i)(f ) | π ∈ Symn}. Hence |Supp(f )| = |{π ∈ Symn |

n

• i=1

mi,π(i)(f ) = 0}|. Thus, we can reduce MS-problem for the Star graph and eigenvalue n − 2 to the problem of finding the minimum value of |{π ∈ Symn |

n

• i=1

mi,π(i) = 0}| for the set of all special n × n matrices M.

Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs

Open problems

1 Characterize all extremal λi(n, 4)-eigenfunctions of H(n, 4) for

i > n

2.

2 MS-problem for the bilinear forms graphs of diameter D ≥ 3. 3 MS-problem for the Doob graphs. Alexandr Valyuzhenich Minimum supports of eigenfunctions of graphs