Nilpotent groups whose Difference graphs have positive genus

Abstract

The power graph of a finite group G is a simple undirected graph with vertex set G and two vertices are adjacent if one is a power of the other. The enhanced power graph of a finite group G is a simple undirected graph whose vertex set is the group G and two vertices a and b are adjacent if there exists c ∈ G such that both a and b are powers of c. In this paper, we study the difference graph D(G) of a finite group G which is the difference of the enhanced power graph and the power graph of G with all isolated vertices removed. We characterize all the finite nilpotent groups G such that the genus (or cross-cap) of the difference graph D(G) is at most 2. 2020 Mathematics Subject Classification. 05C25. Key words and phrases. Enhanced power graph, power graph, nilpotent groups, genus and cross-cap of a graph.