Page 296 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
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GRAPHS AND GROUPS, GEOMETRIES AND GAP - G2G2 (MS-7)
Discrete Fuglede conjecture on cyclic groups
Gergely Kiss, kigergo57@gmail.com
Alfréd Rényi Institute of Mathematics, Hungary
Coauthors: Gábor Somlai, Romanos Diogenes Malikiosis, Máté Vizer
Fuglede in 1974 conjectured that a bounded domain S ⊂ Rd tiles the d-dimensional Euclidean
space if and only if the set of functions in L2(S) admits an orthogonal basis of exponential
functions.
In my talk we focus on the discrete version of Fuglede’s conjecture that can be formulated
as follows. A subset S of a finite abelian group G tiles G if and only if the character table of G
has a submatrix, whose rows are indexed by the elements of S, which is a complex Hadamard
matrix. Fuglede’s original conjecture were disproved first by Tao and the proof is based on a
counterexample on elementary abelian p-groups.
On the other hand, it is still an open question whether the discrete Fuglede’s conjecture is
true on cyclic groups. In my talk I will summarize the known results concerning this question.
In particular, I will present our recent result which shows that the conjecture holds on cyclic
groups whose order is the product of at most 4 (not necessarily different) primes. I will introduce
a geometric technique that we called ’cube-rule’ and which is an essential tool of the proof.
Strongly Deza graphs
Elena V. Konstantinova, e_konsta@math.nsc.ru
Sobolev Institute of Mathematics, Novosibirsk State University, Russian Federation
Coauthors: Saieed Akbari, Vladislav V. Kabanov, Willem H. Haemers,
Mohammad Ali Hosseinzadeh
A Deza graph G with parameters (n, k, b, a) is a k-regular graph of order n for which the number
of common neighbours of two vertices takes just two values, b or a, where b a. Moreover,
G is not the complete graph or the edgeless graph. Deza graphs were introduced in [3], and
the name was given in [4], where the basics of Deza graph theory were founded and different
constructions of Deza graphs were presented. Strongly regular graphs are a particular case of
Deza graphs.
Deza graphs can be considered in terms of matrices. Let G be a graph with n vertices, and
M be its adjacency matrix. Then G is a Deza graph with parameters (n, k, b, a) if and only if
M 2 = aA + bB + kI
for some symmetric (0, 1)-matrices A and B such that A + B + I = J, where J is the all ones
matrix and I is the identity matrix. Graphs GA and GB with matrices A and B are called the
children of G.
Definition. A Deza graph is called a strongly Deza graph if its children are strongly regular
graphs.
Theorem 1. [1, Theorem 3.2] Let G be a Deza graph with parameters (n, k, b, a), b > a. Let
M, A, B be the adjacency matrices of G and its children, respectively. If θ1 = k, θ2, . . . , θn are
the eigenvalues of M , then
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Discrete Fuglede conjecture on cyclic groups
Gergely Kiss, kigergo57@gmail.com
Alfréd Rényi Institute of Mathematics, Hungary
Coauthors: Gábor Somlai, Romanos Diogenes Malikiosis, Máté Vizer
Fuglede in 1974 conjectured that a bounded domain S ⊂ Rd tiles the d-dimensional Euclidean
space if and only if the set of functions in L2(S) admits an orthogonal basis of exponential
functions.
In my talk we focus on the discrete version of Fuglede’s conjecture that can be formulated
as follows. A subset S of a finite abelian group G tiles G if and only if the character table of G
has a submatrix, whose rows are indexed by the elements of S, which is a complex Hadamard
matrix. Fuglede’s original conjecture were disproved first by Tao and the proof is based on a
counterexample on elementary abelian p-groups.
On the other hand, it is still an open question whether the discrete Fuglede’s conjecture is
true on cyclic groups. In my talk I will summarize the known results concerning this question.
In particular, I will present our recent result which shows that the conjecture holds on cyclic
groups whose order is the product of at most 4 (not necessarily different) primes. I will introduce
a geometric technique that we called ’cube-rule’ and which is an essential tool of the proof.
Strongly Deza graphs
Elena V. Konstantinova, e_konsta@math.nsc.ru
Sobolev Institute of Mathematics, Novosibirsk State University, Russian Federation
Coauthors: Saieed Akbari, Vladislav V. Kabanov, Willem H. Haemers,
Mohammad Ali Hosseinzadeh
A Deza graph G with parameters (n, k, b, a) is a k-regular graph of order n for which the number
of common neighbours of two vertices takes just two values, b or a, where b a. Moreover,
G is not the complete graph or the edgeless graph. Deza graphs were introduced in [3], and
the name was given in [4], where the basics of Deza graph theory were founded and different
constructions of Deza graphs were presented. Strongly regular graphs are a particular case of
Deza graphs.
Deza graphs can be considered in terms of matrices. Let G be a graph with n vertices, and
M be its adjacency matrix. Then G is a Deza graph with parameters (n, k, b, a) if and only if
M 2 = aA + bB + kI
for some symmetric (0, 1)-matrices A and B such that A + B + I = J, where J is the all ones
matrix and I is the identity matrix. Graphs GA and GB with matrices A and B are called the
children of G.
Definition. A Deza graph is called a strongly Deza graph if its children are strongly regular
graphs.
Theorem 1. [1, Theorem 3.2] Let G be a Deza graph with parameters (n, k, b, a), b > a. Let
M, A, B be the adjacency matrices of G and its children, respectively. If θ1 = k, θ2, . . . , θn are
the eigenvalues of M , then
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