On gamma-hyperellipticity of graphs - Mednykh, Alexander (Ars mathematica contemporanea, 2016)
The basic objects of research in this paper are graphs and their branched coverings. By a graph, we mean a finite connected multigraph. The genus of a graph is defined as the rank of the first homology group. A graph is said to be ?$γ$?-hyperelliptic if it is a two fold branched covering of a genus ?$γ$ graph?. The corresponding covering involution is called ?$γ$?-hyperelliptic. The aim of the paper is to provide a few criteria for the involution ?$τ$? acting on a graph ?$X$? of genus ?$g$? to be ?$γ$?-hyperelliptic. If $τ$ has at least one fixed point then the first criterium states that there is a basis in the homology group ?$H_1(X)$? whose elements are either invertible or split into ?$γ$? interchangeable pairs under the action of ?$τ_*$?. The second criterium is given by the formula ?$\text{tr}_{\rm{H_1}(X)} (τ_*) = 2 γ - g$?. Similar results are also obtained in the case when ?$τ$? acts fixed point free.
Keywords: graf; graph; hipereliptičen graf; homology group; homološka grupa; hyperelliptic graph; Riemann--Hurwitz formula; Riemann-Hurwitzeva formula; Schreier formula; Schreierjeva formula