Funding: The first author is supported by the PhD fellowship of CSIR (File no. 08/155 (0086)/2020 − EMR − I), Govt. of India. The second author acknowledges the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme Groups, representations and applications: new perspectives (supported by EPSRC grant no. EP/R014604/1), where he held a Simons Fellowship. The third author acknowledges the funding of DST grant SR/F ST/MS − I/2019/41 and MT R/2022/000020, Govt. of India. The fourth author acknowledges SERB-National Post-Doctoral Fellowship (File No. PDF/2021/001899) during the preparation of this work. The difference graph of a finite group D (G) is the difference of the enhanced power graph of G and the power graph of G, where all isolated vertices are removed. In this paper we study the connectedness and perfectness of D (G) with respect to various properties of the underlying group G. We also find several connections between the difference graph of G and the Gruenberg-Kegel graph of G. We also examine the operation of twin reduction on graphs, a technique which produces smaller graphs which may be easier to analyze. Applying this technique to simple groups can have a number of outcomes, not fully understood, but including some graphs with large girth. Peer reviewed