### Abstract

Two particular classes of products are studied in detail: the zig-zag product of a complete graph with a cycle graph, and the zig-zag product of a 4-regular graph with the cycle graph of length 4. Furthermore, an example coming from the theory of Schreier graphs associated with the action of self-similar groups is also considered: the graph products are completely determined and their spectral analysis is developed.

Original language | English |
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Pages (from-to) | 120 |

Number of pages | 151 |

Journal | Journal of graph theory |

Volume | 83 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2016 |

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### Fields of Expertise

- Information, Communication & Computing

### Treatment code (Nähere Zuordnung)

- Theoretical

### Cite this

**Connectedness and isomorphism properties of the zig-zag product of graphs.** / D'Angeli, Daniele; Sava-Huss, Ecaterina; Donno, Alfredo.

Research output: Contribution to journal › Article › Research › peer-review

*Journal of graph theory*, vol. 83, no. 2, pp. 120. https://doi.org/10.1002/jgt.21917

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TY - JOUR

T1 - Connectedness and isomorphism properties of the zig-zag product of graphs

AU - D'Angeli, Daniele

AU - Sava-Huss, Ecaterina

AU - Donno, Alfredo

PY - 2016

Y1 - 2016

N2 - In this paper we investigate the connectedness and the isomorphism problems for zig-zag products of two graphs. A sufficient condition for the zig-zag product of two graphs to be connected is provided, reducing to the study of the connectedness property of a new graph which depends only on the second factor of the graph product. We show that, when the second factor is a cycle graph, the study of the isomorphism problem for the zig-zag product is equivalent to the study of the same problem for the associated pseudo-replacement graph. The latter is defined in a natural way, by a construction generalizing the classical replacement product, and its degree is smaller than the degree of the zig-zag product graph.Two particular classes of products are studied in detail: the zig-zag product of a complete graph with a cycle graph, and the zig-zag product of a 4-regular graph with the cycle graph of length 4. Furthermore, an example coming from the theory of Schreier graphs associated with the action of self-similar groups is also considered: the graph products are completely determined and their spectral analysis is developed.

AB - In this paper we investigate the connectedness and the isomorphism problems for zig-zag products of two graphs. A sufficient condition for the zig-zag product of two graphs to be connected is provided, reducing to the study of the connectedness property of a new graph which depends only on the second factor of the graph product. We show that, when the second factor is a cycle graph, the study of the isomorphism problem for the zig-zag product is equivalent to the study of the same problem for the associated pseudo-replacement graph. The latter is defined in a natural way, by a construction generalizing the classical replacement product, and its degree is smaller than the degree of the zig-zag product graph.Two particular classes of products are studied in detail: the zig-zag product of a complete graph with a cycle graph, and the zig-zag product of a 4-regular graph with the cycle graph of length 4. Furthermore, an example coming from the theory of Schreier graphs associated with the action of self-similar groups is also considered: the graph products are completely determined and their spectral analysis is developed.

UR - http://onlinelibrary.wiley.com/doi/10.1002/jgt.21917/abstract

U2 - 10.1002/jgt.21917

DO - 10.1002/jgt.21917

M3 - Article

VL - 83

SP - 120

JO - Journal of graph theory

JF - Journal of graph theory

SN - 0364-9024

IS - 2

ER -