Conjugates and Factors

2018 ◽  
pp. 109-124
Author(s):  
Christophe Reutenauer
Keyword(s):  
Characteristic Property ◽  
Cayley Graphs ◽  
Cyclic Groups ◽  
Sturmian Words ◽  
Sturmian Sequences ◽  
Burrows Wheeler Transform

Christoffel words are naturally cyclic objects. They may be defined by the Cayley graphs of finite cyclic groups. They have many characterizations through their conjugation classes; among them, one is obtained using the Burrows–Wheeler transform (Mantaci, Restivo, and Sciortino); another one is due to Pirillo. The sets of their circular factors have many remarkable properties; in particular the number of them of length k is k+1, if k is smaller than the length of the Christoffel word, and it is a characteristic property (Borel and the author), reminiscent of the similar property of Sturmian sequences. A related characterization, similar to that of Droubay and Pirillo for Sturmian sequences, rests on the count of palindromic factors. The set of finite Sturmian words, that is, the set of all factors of all Christoffel words, coincides with the set of balanced words (Dulucq and Gouyou–Beauchamps).

scholarly journals Integral Cayley Graphs over Abelian Groups

10.37236/353 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Walter Klotz ◽  
Torsten Sander
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Boolean Algebra ◽  
Cayley Graph ◽  
Cayley Graphs ◽  
Additive Group ◽  
Cyclic Groups ◽  
Vertex Set ◽  
A Finite Group ◽  
Gamma S

Let $\Gamma$ be a finite, additive group, $S \subseteq \Gamma, 0\notin S, -S=\{-s: s\in S\}=S$. The undirected Cayley graph Cay$(\Gamma,S)$ has vertex set $\Gamma$ and edge set $\{\{a,b\}: a,b\in \Gamma$, $a-b \in S\}$. A graph is called integral, if all of its eigenvalues are integers. For an abelian group $\Gamma$ we show that Cay$(\Gamma,S)$ is integral, if $S$ belongs to the Boolean algebra $B(\Gamma)$ generated by the subgroups of $\Gamma$. The converse is proven for cyclic groups. A finite group $\Gamma$ is called Cayley integral, if every undirected Cayley graph over $\Gamma$ is integral. We determine all abelian Cayley integral groups.

Burrows–Wheeler transform and Sturmian words

2003 ◽  
Vol 86 (5) ◽  
pp. 241-246 ◽  
Author(s):  
S. Mantaci ◽  
A. Restivo ◽  
M. Sciortino
Keyword(s):  
Sturmian Words ◽  
Burrows Wheeler Transform

scholarly journals Automorphism Groups of Rational Circulant Graphs

10.37236/2039 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Mikhail Klin ◽  
István Kovács
Keyword(s):  
Orthogonal Group ◽  
Automorphism Groups ◽  
Cayley Graphs ◽  
Symmetric Groups ◽  
Wreath Products ◽  
Circulant Graphs ◽  
Cyclic Groups ◽  
Block Structures

The paper concerns the automorphism groups of Cayley graphs over cyclic groups which have a rational spectrum (rational circulant graphs for short). With the aid of the techniques of Schur rings it is shown that the problem is equivalent to consider the automorphism groups of orthogonal group block structures of cyclic groups. Using this observation, the required groups are expressed in terms of generalized wreath products of symmetric groups.

scholarly journals Integral Cayley Graphs Defined by Greatest Common Divisors

10.37236/581 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Walter Klotz ◽  
Torsten Sander
Keyword(s):  
Abelian Group ◽  
Boolean Algebra ◽  
Cayley Graph ◽  
Direct Product ◽  
Undirected Graph ◽  
Cayley Graphs ◽  
Cyclic Groups ◽  
Greatest Common Divisors ◽  
Gamma S

An undirected graph is called integral, if all of its eigenvalues are integers. Let $\Gamma =Z_{m_1}\otimes \ldots \otimes Z_{m_r}$ be an abelian group represented as the direct product of cyclic groups $Z_{m_i}$ of order $m_i$ such that all greatest common divisors $\gcd(m_i,m_j)\leq 2$ for $i\neq j$. We prove that a Cayley graph $Cay(\Gamma,S)$ over $\Gamma$ is integral, if and only if $S\subseteq \Gamma$ belongs to the the Boolean algebra $B(\Gamma)$ generated by the subgroups of $\Gamma$. It is also shown that every $S\in B(\Gamma)$ can be characterized by greatest common divisors.

scholarly journals A Note on Constructing Large Cayley Graphs of Given Degree and Diameter by Voltage Assignments

10.37236/1347 ◽  
1997 ◽  
Vol 5 (1) ◽  
Author(s):  
Ljiljana Branković ◽  
Mirka Miller ◽  
Ján Plesník ◽  
Joe Ryan ◽  
Jozef Širáň
Keyword(s):  
Cayley Graphs ◽  
Computer Search ◽  
Large Graphs ◽  
Covering Spaces ◽  
Semidirect Products ◽  
Cyclic Groups ◽  
Small Base ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs

Voltage graphs are a powerful tool for constructing large graphs (called lifts) with prescribed properties as covering spaces of small base graphs. This makes them suitable for application to the degree/diameter problem, which is to determine the largest order of a graph with given degree and diameter. Many currently known largest graphs of degree $\le 15$ and diameter $\le 10$ have been found by computer search among Cayley graphs of semidirect products of cyclic groups. We show that all of them can in fact be described as lifts of smaller Cayley graphs of cyclic groups, with voltages in (other) cyclic groups. This opens up a new possible direction in the search for large vertex-transitive graphs of given degree and diameter.

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