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, b are adjacent if there exists \(c \in G\) such that both a and b are powers of c. In this paper, we study the difference graph \(\mathcal {D}(G)\) 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 \(\mathcal {D}(G)\) is at most 2.