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| Main Authors: | , , |
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| Format: | Preprint |
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2020
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| Online Access: | https://arxiv.org/abs/2005.05422 |
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| _version_ | 1866912145950638080 |
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| author | Miklavič, Štefko Šparl, Primož Wilson, Stephen E. |
| author_facet | Miklavič, Štefko Šparl, Primož Wilson, Stephen E. |
| contents | A subgroup of the automorphism group of a graph acts {\em half-arc-transitively} on the graph if it acts transitively on the vertex-set and on the edge-set of the graph but not on the arc-set of the graph. If the full automorphism group of the graph acts half-arc-transitively, the graph is said to be {\em half-arc-transitive}.
In 1994 Gardiner and Praeger introduced two families of tetravalent arc-transitive graphs, called the $C^{\pm 1}$ and the $C^{\pm \varepsilon}$ graphs, that play a prominent role in the characterization of the tetravalent graphs admitting an arc-transitive group of automorphisms with a normal elementary abelian subgroup such that the corresponding quotient graph is a cycle. All of the Gardiner-Praeger graphs are arc-transitive but admit a half-arc-transitive group of automorphisms. Quite recently, Poto\v cnik and Wilson introduced the family of CPM graphs, which are generalizations of the Gardiner-Praeger graphs. Most of these graphs are arc-transitive, but some of them are half-arc-transitive. In fact, at least up to order $1000$, each tetravalent half-arc-transitive loosely-attached graph of odd radius having vertex-stabilizers of order greater than $2$ is isomorphic to a CPM graph.
In this paper we determine the automorphism group of the CPM graphs and investigate isomorphisms between them. Moreover, we determine which of these graphs are $2$-arc-transitive, which are arc-transitive but not $2$-arc-transitive, and which are half-arc-transitive. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2005_05422 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Generalized Gardiner-Praeger graphs and their symmetries Miklavič, Štefko Šparl, Primož Wilson, Stephen E. Combinatorics 20B25 A subgroup of the automorphism group of a graph acts {\em half-arc-transitively} on the graph if it acts transitively on the vertex-set and on the edge-set of the graph but not on the arc-set of the graph. If the full automorphism group of the graph acts half-arc-transitively, the graph is said to be {\em half-arc-transitive}. In 1994 Gardiner and Praeger introduced two families of tetravalent arc-transitive graphs, called the $C^{\pm 1}$ and the $C^{\pm \varepsilon}$ graphs, that play a prominent role in the characterization of the tetravalent graphs admitting an arc-transitive group of automorphisms with a normal elementary abelian subgroup such that the corresponding quotient graph is a cycle. All of the Gardiner-Praeger graphs are arc-transitive but admit a half-arc-transitive group of automorphisms. Quite recently, Poto\v cnik and Wilson introduced the family of CPM graphs, which are generalizations of the Gardiner-Praeger graphs. Most of these graphs are arc-transitive, but some of them are half-arc-transitive. In fact, at least up to order $1000$, each tetravalent half-arc-transitive loosely-attached graph of odd radius having vertex-stabilizers of order greater than $2$ is isomorphic to a CPM graph. In this paper we determine the automorphism group of the CPM graphs and investigate isomorphisms between them. Moreover, we determine which of these graphs are $2$-arc-transitive, which are arc-transitive but not $2$-arc-transitive, and which are half-arc-transitive. |
| title | Generalized Gardiner-Praeger graphs and their symmetries |
| topic | Combinatorics 20B25 |
| url | https://arxiv.org/abs/2005.05422 |