Regular Steinhaus Graphs
Of Odd Degree
Abstract
A Steinhaus matrix is a binary square matrix of size which is symmetric, with diagonal of zeros, and whose uppertriangular coefficients satisfy for all . Steinhaus matrices are determined by their first row. A Steinhaus graph is a simple graph whose adjacency matrix is a Steinhaus matrix. We give a short new proof of a theorem, due to Dymacek, which states that even Steinhaus graphs, i.e. those with all vertex degrees even, have doublysymmetric Steinhaus matrices. In 1979 Dymacek conjectured that the complete graph on two vertices is the only regular Steinhaus graph of odd degree. Using Dymacek’s theorem, we prove that if is a Steinhaus matrix associated with a regular Steinhaus graph of odd degree then its submatrix is a multisymmetric matrix, that is a doublysymmetric matrix where each row of its uppertriangular part is a symmetric sequence. We prove that the multisymmetric Steinhaus matrices of size whose Steinhaus graphs are regular modulo , i.e. where all vertex degrees are equal modulo , only depend on parameters for all even numbers , and on parameters in the odd case. This result permits us to verify the Dymacek’s conjecture up to vertices in the odd case.
1 Introduction
Let be a binary sequence of length with entries in . The Steinhaus matrix associated with is the square matrix of size , defined as follows:

for all ,

for all ,

for all ,

for all .
By convention is the Steinhaus matrix of size associated with the empty sequence. For example, the following matrix in is the Steinhaus matrix associated with the binary sequence of length .
The set of all Steinhaus matrices of size will be denoted by . It is clear that, for every positive integer , the set has a cardinality of .
The Steinhaus triangle associated with is the uppertriangular part of the Steinhaus matrix . It was introduced by Hugo Steinhaus in 1963 [Steinhaus1963], who asked whether there exists a Steinhaus triangle containing as many ’s as ’s for each admissible size. Solutions of this problem appeared in [Harborth1972, Eliahou2004]. A generalization of this problem to all finite cyclic groups was posed in [Molluzzo1978] and was partially solved in [Chappelon2008].
The Steinhaus graph associated with is the simple graph on vertices whose adjacency matrix is the Steinhaus matrix . A vertex of a Steinhaus graph is usually labelled by its corresponding row number in and the th vertex of will be denoted by . For instance, the following graph is the Steinhaus graph associated with the sequence .
For every positive integer , the zeroedge graph on vertices is the Steinhaus graph associated with the sequence of zeros of length .
Steinhaus graphs were introduced by Molluzzo in 1978 [Molluzzo1978]. A general problem on Steinhaus graphs is that of characterizing those satisfying a given graph property. The bipartite Steinhaus graphs were characterized in [Chang1999, Dymacek1986, Dymacek1995] and the planar ones in [Dymacek2000a]. In [Dymacek2000], the following conjectures were made:
Conjecture 1.
The regular Steinhaus graphs of even degree are the zeroedge graph on vertices, for all positive integers , and the Steinhaus graph on vertices generated by the periodic sequence of length , for all positive integers .
Conjecture 2.
The complete graph on two vertices is the only regular Steinhaus graph of odd degree.
These conjectures were verified up to in 1988 by exhaustive search [Bailey1988]. More recently [Augier2008], Augier and Eliahou extended the verification up to vertices by considering the weaker notion of parityregular Steinhaus graphs, i.e. Steinhaus graphs where all vertex degrees have the same parity. They searched regular graphs in the set of parityregular Steinhaus graphs. This has enabled them to perform the verification because it is known that Steinhaus matrices associated with parityregular Steinhaus graphs on vertices depend on approximately parameters [Bailey1988, Augier2008]. This result is based on a theorem, due to Dymacek, which states that Steinhaus matrices associated with parityregular Steinhaus graphs of even type are doublysymmetric matrices, i.e. where all the entries are symmetric with respect to the diagonal and the antidiagonal of the matrices. A short new proof of this theorem is given in Section . Using Dymacek’s theorem, Bailey and Dymacek showed [Bailey1988] that binary sequences associated with regular Steinhaus graphs of odd degree are of the form . In Section , we refine this result and, more precisely, we prove that if is a Steinhaus matrix associated with a regular Steinhaus graph of odd degree, then its submatrix is a multisymmetric Steinhaus matrix, i.e. a doublysymmetric matrix where each row of the uppertriangular part is a symmetric sequence. A parametrization and a counting of multisymmetric Steinhaus matrices of size are also given in Section for all . In Section , we show that, for a Steinhaus graph whose Steinhaus matrix is multisymmetric, the knowledge of the vertex degrees modulo leads to a system of binary equations on the entries of its Steinhaus matrix. In Section , we study the special case of multisymmetric Steinhaus matrices whose Steinhaus graphs are regular modulo , i.e. where all vertex degrees are equal modulo . We show that a such matrix of size only depends on parameters for all even, and on parameters in the odd case. Using these parametrizations, we obtain, by computer search, that for all positive integers , the zeroedge graph on vertices is the only Steinhaus graph on vertices with a multisymmetric matrix and which is regular modulo . This permits us to extend the verification of Conjecture 2 up to vertices.
2 A new proof of Dymacek’s theorem
Recall that a square matrix of size is said to be doublysymmetric if the entries of are symmetric with respect to the diagonal and to the antidiagonal of , that is
In [Dymacek2000], Dymacek characterized the parityregular Steinhaus graphs. These results are based on the following theorem on parityregular Steinhaus graphs of even type, where all vertex degrees are even.
Theorem 2.1 (Dymacek’s theorem).
The Steinhaus matrix of a parityregular Steinhaus graph of even type is doublysymmetric.
In this section we give a new easier proof of Dymacek’s theorem. The main idea of our proof is that the antidiagonal entries of a Steinhaus matrix are determined by the vertex degrees of its associated Steinhaus graph.
Theorem 2.2.
Let be a Steinhaus graph on vertices and its associated Steinhaus matrix. Then every antidiagonal entry of can be expressed by means of the vertex degrees of . If we denote by the degree of the vertex in , then for all , we have
The proof is based on the following lemma which shows that each entry of the uppertriangular part of a Steinhaus matrix can be expressed by means of the entries of the first row , the last column or the overdiagonal of .
Lemma 2.3.
Let be a Steinhaus matrix of size . Then, for all , we have
Proof.
Easily follows from the relation: for all . ∎
Proof of Theorem 2.2.
We begin by expressing each vertex degree of the Steinhaus graph by means of the entries of the first row, the last column and the overdiagonal of . Here we view the entries as , integers. For all , we obtain
By Lemma 2.3, it follows that
for all . The second congruence can be treated by the same way. ∎
Remark.
We deduce from Theorem 2.2 a necessary condition on the vertex degrees of a given labelled graph to be a Steinhaus graph. Indeed, vertex degrees of a Steinhaus graph on vertices must satisfy the following binary equations:
More generally, an open problem, corresponding to Question in [Dymacek1996], is to determine if an arbitrary graph, not necessary labelled, is isomorphic to a Steinhaus graph.
Now, we characterize doublysymmetric Steinhaus matrices.
Proposition 2.4.
Let be a Steinhaus matrix of size . Then the following assertions are equivalent:

the matrix is doublysymmetric,

the overdiagonal of is a symmetric sequence,

the entries of the antidiagonal of vanish for all .
Proof.
Trivial.
Suppose that the overdiagonal of is a symmetric sequence, that is
for all . If is odd, then we have
for all . Otherwise, if is even, then we obtain
for all .
By induction on . Consider the submatrix that is a Steinhaus matrix of size . By induction hypothesis, the matrix is doublysymmetric. Then it remains to prove that for all . First, since , it follows that and for all , we have
∎
We are now ready to prove Dymacek’s theorem.
3 Multisymmetric Steinhaus matrices
In this section, we will study in detail the structure of Steinhaus matrices associated with regular Steinhaus graphs of odd degree.
Let be a Steinhaus graph on vertices. Then, for every integer , we denote by the graph obtained from by deleting its th vertex and its incident edges in . Since the adjacency matrix of the graph (resp. ) is the Steinhaus matrix obtained by removing the first row (resp. the last column) in the adjacency matrix of , it follows that the graph (resp. ) is a Steinhaus graph on vertices.
Bailey and Dymacek studied the regular Steinhaus graphs of odd degree in [Bailey1988], where the following theorem is stated, using Dymacek’s theorem.
Theorem 3.1 ([Bailey1988]).
Let be a regular Steinhaus graph of odd degree on vertices. Then , the Steinhaus graph is regular of even degree , and for all .
Remark.
In every simple graph, there are an even number of vertices of odd degree. Therefore parityregular Steinhaus graphs of odd type and thus regular Steinhaus graphs of odd degree have an even number of vertices.
In their theorem, the authors studied the form of the sequence associated with . We are more interested in the Steinhaus matrix of in the sequel.
Recall that a square matrix of size is said to be multisymmetric if is doublysymmetric and each row of the uppertriangular part of is a symmetric sequence, that is
First, it is easy to see that each column of the uppertriangular part of a multisymmetric matrix is also a symmetric sequence.
Proposition 3.2.
Let be a multisymmetric matrix of size . Then, each column of the uppertriangular part of is a symmetric sequence, that is for all .
Proof.
Easily follows from the relation: for all . ∎
As for doublysymmetric Steinhaus matrices, multisymmetric Steinhaus matrices can be characterized as follows.
Proposition 3.3.
Let be a Steinhaus matrix of size . Then the following assertions are equivalent:

the matrix is multisymmetric,

the first row, the last column and the overdiagonal of are symmetric sequences,

the entries , and vanish for all .
We now refine Theorem 3.1.
Theorem 3.4.
Let be a regular Steinhaus graph of odd degree on vertices. Then is a regular Steinhaus graph of even degree whose associated Steinhaus matrix is multisymmetric.
Proof.
Let be the Steinhaus matrix associated with . Theorem 3.1 implies that the Steinhaus graph is regular of even degree and that we have
for all . Therefore, for all , we have
Then the first row of the matrix , the Steinhaus matrix of the graph , is a symmetric sequence. Moreover, by Dymacek’s theorem, the matrix is doublysymmetric. Finally, by Proposition 3.3, the matrix is multisymmetric. ∎
Remark.
By Theorem 3.4, it is easy to show that Conjecture 1 implies Conjecture 2. Indeed, if Conjecture 1 is true, then the zeroedge graph on vertices is the only regular Steinhaus graph of even degree whose Steinhaus matrix is multisymmetric. It follows, by Theorem 3.4, that if is a regular Steinhaus graph of odd degree on vertices then or . Therefore the Steinhaus graph is the star graph on vertices which is not a regular Steinhaus graph.
In the sequel of this section we will study in detail the multisymmetric Steinhaus matrices. First, in order to determine a parametrization of these matrices, we introduce the following operator
which assigns to each matrix in the Steinhaus matrix in defined by , for all . As depicted in the following matrix, the uppertriangular part of is an extension of the uppertriangular part of .
Proposition 3.5.
Let be a Steinhaus matrix of size . Then the extension of only depends on the parameters , and , with in .
Proof.
Let . Each entry , for , can be expressed by means of and the entries of . Indeed, we have
Then the entries , and determine the extension of . ∎
Therefore, for every Steinhaus matrix of size , there exist distinct Steinhaus matrices of size such that . We can also use this operator to determine parametrizations of multisymmetric Steinhaus matrices.
Proposition 3.6.
Let be a multisymmetric Steinhaus matrix of size . Let be an element of the set for all . Then the matrix depends on the following parameters:

and , for even,

, for odd.
Proof.
For all positive integers , the number of multisymmetric Steinhaus matrices of size immediately follows.
Theorem 3.7.
Let be a positive integer. If we denote by the number of multisymmetric Steinhaus matrices of size , then we have
4 Vertex degrees of Steinhaus graphs associated with
multisymmetric Steinhaus matrices
In this section, we analyse the vertex degrees of a Steinhaus graph associated with a multisymmetric Steinhaus matrix of size . We begin with the case of doublysymmetric Steinhaus matrices.
Proposition 4.1.
Let be a positive integer and be a Steinhaus graph on vertices whose Steinhaus matrix is doublysymmetric. Then, for all , we have
Proof.
If we denote by the Steinhaus matrix associated with the graph , then, for all , we have
∎
We shall now see that, for a Steinhaus graph associated with a multisymmetric Steinhaus matrix, the knowledge of the vertex degrees modulo imposes strong conditions on the entries of its Steinhaus matrix. In order to prove this result, we distinguish different cases depending on the parity of .
Proposition 4.2.
Let be an even number and be a Steinhaus graph on vertices whose Steinhaus matrix is multisymmetric. Then, we have
Proof.
Remark.
Let be an even number. In every Steinhaus graph on vertices whose Steinhaus matrix is multisymmetric the fourth vertex has a degree divisible by .
Proposition 4.3.
Let be an odd number and be a Steinhaus graph on vertices whose Steinhaus matrix is multisymmetric. Then, we have
Proof.
Remark.
Let be an odd number. In every Steinhaus graph on vertices whose Steinhaus matrix is multisymmetric the third vertex has a degree divisible by .
5 Multisymmetric Steinhaus matrices
of Steinhaus graphs with regularity modulo
In this section, we consider the multisymmetric Steinhaus matrices associated with Steinhaus graphs which are regular modulo , i.e. where all vertex degrees are equal modulo . First, we determine an upper bound of the number of these matrices. Two cases are distinguished, according to the parity of .
Theorem 5.1.
For all odd numbers , there are at most multisymmetric Steinhaus matrices of size whose associated Steinhaus graphs are regular modulo .
Proof.
Let be an odd number and a multisymmetric Steinhaus matrix of size . By Proposition 3.6, the matrix depends on the parameters for . If the Steinhaus graph associated with is regular modulo , then Proposition 4.3 implies that for all and thus
for all .
If , then is odd and
for all . Therefore the matrix can be parametrized by
with
Suppose that we know the parameters in
Then, by Proposition 3.6 again, the multisymmetric matrix can be parametrized by . Therefore the entries
in depend on the parameters in . Moreover, if the Steinhaus graph associated with is regular modulo , then Proposition 4.3 implies that
for all . If the inequality
holds, then the entries depend on the parameters in for all . Since we have for all , it follows that the entries
depend on the parameters in . Suppose now that is solution of the following inequality
Therefore the extension of depends on the entries for and which vanishes by Proposition 4.3. Thus, all the entries of the matrix depend on the parameters in . Finally, a solution of this inequality can be obtained when
If , then is even and
for all . Therefore the matrix can be parametrized by
with
As above, in the case , we can prove that all the entries of the matrix depend on the parameters in
if is solution of the following inequality
A solution is obtained when
∎
Theorem 5.2.
For all even numbers , there are at most multisymmetric Steinhaus matrices of size whose associated Steinhaus graphs are regular modulo .
Sketch of proof.
Similar to the proof of Theorem 5.1. Let be a multisymmetric Steinhaus matrix of even size . First, by Proposition 3.6, for all positive integers with , the multisymmetric Steinhaus matrix can be parametrized by the entries in
Moreover, if the Steinhaus graph associated with is regular modulo , then Proposition 4.2 implies that and for all . It follows that the entries also depends on the parameters in for all . Finally, we can see that, if is solution of the following inequality
then, as in the proof of Proposition 3.6, the extension of depends on the entries for