On the power graphs of elementary abelian and extra special $p$-groups

Abstract

‎For a given odd prime $p$‎, ‎we‎ ‎investigate the power graphs of three‎ ‎classes of finite groups‎: ‎the elementary‎ ‎abelian groups of exponent $p$‎, ‎and the‎ ‎extra special groups of exponents $p$ or‎ ‎$p^2$‎. ‎We show that these power graphs‎ ‎are Eulerian for every $p$‎. ‎As a‎ ‎corollary‎, ‎we describe two classes of‎ ‎non-isomorphic groups with isomorphic‎ ‎power graphs‎. ‎In addition‎, ‎we prove that‎ ‎the clique graphs of the power graphs of‎ ‎two considered classes are complete‎.