On two-sided group digraphs and graphs

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 (2S) over right arrow (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(-1) xr for some l is an element of L and r is an element of 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 (2S) over right arrow (G; L, R) is equal to the number of double cosets of the pair L, R when they are two subgroups of G.