Geodesic
Fixed points of cyclic groups acting purely harmonically on a graph
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 18 (2021) no. 1, pp. 617-621

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Let $X$ be a finite connected graph, possibly with loops and multiple edges. An automorphism group of $X$ acts purely harmonically if it acts freely on the set of directed edges of $X$ and has no invertible edges. Define a genus $g$ of the graph $X$ to be the rank of the first homology group. A discrete version of the Wiman theorem states that the order of a cyclic group $\mathbb{Z}_n$ acting purely harmonically on a graph $X$ of genus $g>1$ is bounded from above by $2g+2.$ In the present paper, we investigate how many fixed points has an automorphism generating a «large» cyclic group $\mathbb{Z}_n$ of order $n\ge2g-1.$ We show that in the most cases, the automorphism acts fixed point free, while for groups of order $2g$ and $2g-1$ it can have one or two fixed points.
Keywords: graph, fixed point, Wiman theorem.
Mots-clés : homological genus, harmonic automorphism 
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     author = {A. D. Mednykh},
     title = {Fixed points of cyclic groups acting purely harmonically on a graph},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {617--621},
     publisher = {mathdoc},
     volume = {18},
     number = {1},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2021_18_1_a19/}
}
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A. D. Mednykh. Fixed points of cyclic groups acting purely harmonically on a graph. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 18 (2021) no. 1, pp. 617-621. http://geodesic.mathdoc.fr/item/SEMR_2021_18_1_a19/