REGULAR STEINHAUS GRAPHSOF ODD DEGREE

Regular Steinhaus Graphs
Of Odd Degree

Jonathan Chappelon
02/09/2009
Abstract

A Steinhaus matrix is a binary square matrix of size which is symmetric, with diagonal of zeros, and whose upper-triangular 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 doubly-symmetric 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 sub-matrix is a multi-symmetric matrix, that is a doubly-symmetric matrix where each row of its upper-triangular part is a symmetric sequence. We prove that the multi-symmetric 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 upper-triangular 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 zero-edge 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 zero-edge 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 parity-regular Steinhaus graphs, i.e. Steinhaus graphs where all vertex degrees have the same parity. They searched regular graphs in the set of parity-regular Steinhaus graphs. This has enabled them to perform the verification because it is known that Steinhaus matrices associated with parity-regular 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 parity-regular Steinhaus graphs of even type are doubly-symmetric matrices, i.e. where all the entries are symmetric with respect to the diagonal and the anti-diagonal 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 sub-matrix is a multi-symmetric Steinhaus matrix, i.e. a doubly-symmetric matrix where each row of the upper-triangular part is a symmetric sequence. A parametrization and a counting of multi-symmetric Steinhaus matrices of size are also given in Section for all . In Section , we show that, for a Steinhaus graph whose Steinhaus matrix is multi-symmetric, 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 multi-symmetric 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 zero-edge graph on vertices is the only Steinhaus graph on vertices with a multi-symmetric 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 doubly-symmetric if the entries of are symmetric with respect to the diagonal and to the anti-diagonal of , that is

In [Dymacek2000], Dymacek characterized the parity-regular Steinhaus graphs. These results are based on the following theorem on parity-regular Steinhaus graphs of even type, where all vertex degrees are even.

Theorem 2.1 (Dymacek’s theorem).

The Steinhaus matrix of a parity-regular Steinhaus graph of even type is doubly-symmetric.

In this section we give a new easier proof of Dymacek’s theorem. The main idea of our proof is that the anti-diagonal 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 anti-diagonal 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 upper-triangular part of a Steinhaus matrix can be expressed by means of the entries of the first row , the last column or the over-diagonal 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 over-diagonal 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 doubly-symmetric Steinhaus matrices.

Proposition 2.4.

Let be a Steinhaus matrix of size . Then the following assertions are equivalent:

  • the matrix is doubly-symmetric,

  • the over-diagonal of is a symmetric sequence,

  • the entries of the anti-diagonal of vanish for all .

Proof.

Trivial.
Suppose that the over-diagonal 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 sub-matrix that is a Steinhaus matrix of size . By induction hypothesis, the matrix is doubly-symmetric. 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.

Proof of Theorem 2.1.

Let be a parity-regular Steinhaus graph of even type on vertices and its Steinhaus matrix. If , then which is trivially doubly-symmetric. Otherwise, for , Theorem 2.2 implies that

for all . Finally, the matrix is doubly-symmetric by Proposition 2.4. ∎

3 Multi-symmetric 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 parity-regular 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 multi-symmetric if is doubly-symmetric and each row of the upper-triangular part of is a symmetric sequence, that is

First, it is easy to see that each column of the upper-triangular part of a multi-symmetric matrix is also a symmetric sequence.

Proposition 3.2.

Let be a multi-symmetric matrix of size . Then, each column of the upper-triangular part of is a symmetric sequence, that is for all .

Proof.

Easily follows from the relation: for all . ∎

As for doubly-symmetric Steinhaus matrices, multi-symmetric 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 multi-symmetric,

  • the first row, the last column and the over-diagonal of are symmetric sequences,

  • the entries , and vanish for all .

Proof.

Similar to the proof of Proposition 2.4 and by using Lemma 2.3 and Proposition 3.2. ∎

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 multi-symmetric.

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 doubly-symmetric. Finally, by Proposition 3.3, the matrix is multi-symmetric. ∎

Remark.

By Theorem 3.4, it is easy to show that Conjecture 1 implies Conjecture 2. Indeed, if Conjecture 1 is true, then the zero-edge graph on vertices is the only regular Steinhaus graph of even degree whose Steinhaus matrix is multi-symmetric. 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 multi-symmetric 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 upper-triangular part of is an extension of the upper-triangular 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 multi-symmetric Steinhaus matrices.

Proposition 3.6.

Let be a multi-symmetric 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.

Let be a multi-symmetric matrix of size . We consider the sub-matrices . By successive application of Proposition 3.5 on the extension of and since the entries , and vanish for all by Proposition 3.3, the parametrizations of the multi-symmetric matrix follow. ∎

For all positive integers , the number of multi-symmetric Steinhaus matrices of size immediately follows.

Theorem 3.7.

Let be a positive integer. If we denote by the number of multi-symmetric Steinhaus matrices of size , then we have

4 Vertex degrees of Steinhaus graphs associated with
multi-symmetric Steinhaus matrices

In this section, we analyse the vertex degrees of a Steinhaus graph associated with a multi-symmetric Steinhaus matrix of size . We begin with the case of doubly-symmetric Steinhaus matrices.

Proposition 4.1.

Let be a positive integer and be a Steinhaus graph on vertices whose Steinhaus matrix is doubly-symmetric. 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 multi-symmetric 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 multi-symmetric. Then, we have

Proof.

First, Proposition 3.3 implies that the entries and vanish for all . This leads to

and, for all , we have

Finally, we complete the proof by Proposition 4.1. ∎

Remark.

Let be an even number. In every Steinhaus graph on vertices whose Steinhaus matrix is multi-symmetric 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 multi-symmetric. Then, we have

Proof.

Proposition 3.3 implies that the entries and vanish for all . Since each row and each column of the upper triangular part of is symmetric, we can use the relation

as in the proof of Proposition 4.2, and the results follow. ∎

Remark.

Let be an odd number. In every Steinhaus graph on vertices whose Steinhaus matrix is multi-symmetric the third vertex has a degree divisible by .

5 Multi-symmetric Steinhaus matrices
of Steinhaus graphs with regularity modulo

In this section, we consider the multi-symmetric 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 multi-symmetric Steinhaus matrices of size whose associated Steinhaus graphs are regular modulo .

Proof.

Let be an odd number and a multi-symmetric 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 multi-symmetric 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 multi-symmetric 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 multi-symmetric Steinhaus matrix of even size . First, by Proposition 3.6, for all positive integers with , the multi-symmetric 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