The power graph of a group G (denoted by P(G)) is the graph whose vertex set is G and two distinct vertices are adjacent if one is the power of the other. By removing the identity from a group’s power graph, P(G), one can obtain the group’s proper power graph. A graph is self-complementary if it is isomorphic to its complement. In this paper, we observe that if G is a p-group, then \(P^*(G)\) is never self-complementary. For the case of an EPPO group G, \(P^*(G)\) is not self-complementary. Moreover, we show that if G is a group having two or more distinct prime divisors, then there does not exist any finite group G except possibly \(G\ncong C_{n}\rtimes C_{p_{k}^{\alpha _{k}}}\) with \(p_{k} \not \mid n\) and \(C_{n}\rtimes Q_{2^r}\) with \(2\not \mid n\) and \(Q_{2^r}\) is the generalized quarternion group such that its proper power graph is self-complementary. The self-complementary index of a graph \(\Gamma \) , denoted by \(s(\Gamma )\) , is a graph parameter that measures the closeness of a graph to being self-complementary. In this paper, we provide suitable sharp bounds (upper and lower) for the self-complementary index of \(P^*(G)\) . Besides that, we look into the finite groups whose power graphs are self-complementary in the sense of different types of forbidden subgraphs and graph theory properties such as regularity, unicyclicity, 2-connectedness, strongly regularity, etc.
