On two-sided group digraphs and graphs

Abstract

In this paper, we consider a generalization of Cayley graphs and digraphs (directed graphs) introduced by Iradmusa and Praeger.  For non-empty subsets L, R of group G, two-sided group digraph \(\overrightarrow{2S}\)(G; L, R) has been defined as a digraph having the vertex set G, and an arc from x to y if and only if y = l⁻¹xr for some l ∈ L and r ∈ R.  This article has strived to answer some open problems posed by Iradmusa and Praeger related to these graphs.  Further, we determine sufficient conditions by which two-sided group graphs to be non-planar, and then we consider some specific cases on subsets L, R.  We prove that the number of connected components of \(\overrightarrow{2S}\)(G; L, R) is equal to the number of double cosets of the pair L, R when they are two subgroups of G.