The relative fixity of a permutation group is the maximum proportion of the points fixed by a non-trivial element of the group, and the relative fixity of a graph is the relative fixity of its automorphism group, viewed as a permutation group on the vertex-set of the graph. We prove in this paper that the relative fixity of connected 2-arc-transitive graphs of a fixed valence tends to 0 as the number of vertices grows to infinity. We prove the same result for the class of arc-transitive graphs of a fixed prime valence, and more generally, for any class of arc-transitive locally-L graphs, where L is a fixed quasiprimitive graph-restrictive permutation group.

Lehner, F., Potocnik, P., Spiga, P. (2021). On fixity of arc-transitive graphs. SCIENCE CHINA. MATHEMATICS, 64(12), 2603-2610 [10.1007/s11425-020-1825-1].

### On fixity of arc-transitive graphs

#### Abstract

The relative fixity of a permutation group is the maximum proportion of the points fixed by a non-trivial element of the group, and the relative fixity of a graph is the relative fixity of its automorphism group, viewed as a permutation group on the vertex-set of the graph. We prove in this paper that the relative fixity of connected 2-arc-transitive graphs of a fixed valence tends to 0 as the number of vertices grows to infinity. We prove the same result for the class of arc-transitive graphs of a fixed prime valence, and more generally, for any class of arc-transitive locally-L graphs, where L is a fixed quasiprimitive graph-restrictive permutation group.
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20B25; arc-transitive; automorphism group; fixed points; fixity; graph; minimal degree; permutation group; vertex-transitive;
English
2021
64
12
2603
2610
open
Lehner, F., Potocnik, P., Spiga, P. (2021). On fixity of arc-transitive graphs. SCIENCE CHINA. MATHEMATICS, 64(12), 2603-2610 [10.1007/s11425-020-1825-1].
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