# Cayley Digraphs of Finite Abelian Groups and Monomial Ideals

TL;DR: The description of the diagrams known as L-shapes provides an efficient method for computing the diameter and the average minimum distance of the corresponding graphs and presents a family of Cayley digraphs parametrized by their diameter, all of them associated to irreducible monomial ideals.

Abstract: In the study of double-loop computer networks, the diagrams known as L-shapes arise as a graphical representation of an optimal routing for every graph's node. The description of these diagrams provides an efficient method for computing the diameter and the average minimum distance of the corresponding graphs. We extend these diagrams to multiloop computer networks. For each Cayley digraph with a finite abelian group as vertex set, we define a monomial ideal and consider its representations via its minimal system of generators or its irredundant irreducible decomposition. From this last piece of information, we can compute the graph's diameter and average minimum distance. That monomial ideal is the initial ideal of a certain lattice with respect to a graded monomial ordering. This result permits the use of Grobner bases for computing the ideal and finding an optimal routing. Finally, we present a family of Cayley digraphs parametrized by their diameter $d$, all of them associated to irreducible monomial ideals.

## Summary (1 min read)

### 1. Introduction.

- In section 2 the authors collect several known facts about monomial ideals, presenting examples and fixing notation for later use.
- Section 6 presents an algorithm specifically tailored for degree three circulants.
- Section 7 is dedicated to providing formulae to find the diameter and the average minimum distance.

### 9. Conclusions.

- In this paper the authors have proposed monomial ideals as a natural tool for studying Cayley digraphs with a finite abelian group as vertex set.
- The authors have generalized the L-shape concept in the plane to L-shape in the r-dimensional affine space.
- From a more practical point of view, it would be interesting to investigate the implementation in computer networks of the family of circulant graphs of degree three under parameters such as routing, fault tolerance, etc. Downloaded 02/05/13 to 193.144.185.28.
- Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php.

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### Cites methods from "Cayley Digraphs of Finite Abelian G..."

...Another method to compute a shortest path using Groebner Basis is presented in papers [16,19]....

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### Cites methods from "Cayley Digraphs of Finite Abelian G..."

...Although L-shape [1], [20] for double-loop networks has been regarded as an important tool for studying the diameter and distance properties of double-loop networks, it is difficult to use L-shape to study the embedding problems for regular graphs on double-loop networks due to the asymmetry of L-shape....

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##### References

6,747 citations

3,634 citations

### "Cayley Digraphs of Finite Abelian G..." refers methods in this paper

...This object has been used to solve many problems in mathematics and computer science (see, for instance, [4, 16, 24, 26])....

[...]

3,248 citations

3,207 citations

### "Cayley Digraphs of Finite Abelian G..." refers methods in this paper

...This object has been used to solve many problems in mathematics and computer science (see, for instance, [4, 16, 24, 26])....

[...]

1,733 citations

### "Cayley Digraphs of Finite Abelian G..." refers background in this paper

...There are different ways to relate monomial ideals with graphs (see, for instance, [30])....

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...Here we review several basic related results and definitions concerning monomial ideals; see, for instance, [2, 30]....

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...The theory of Gröbner bases is related to several areas in mathematics and computer science; see, for instance, [2, 17, 30]....

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