Let [Formula: see text] be a group. The enhanced power graph of [Formula: see text] is a graph with vertex set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if there exists [Formula: see text] such that [Formula: see text] and [Formula: see text] for some [Formula: see text]. Also, a vertex of a graph is called dominating vertex if it is adjacent to every other vertex of the vertex set. Moreover, an enhanced power graph is said to be a dominatable graph if it has a dominating vertex other than the identity element. In an article of 2018, Bera and his coauthor characterized all abelian finite groups and nonabelian finite [Formula: see text]-groups such that their enhanced power graphs are dominatable (see [2]). In addition as an open problem, they suggested characterizing all finite nonabelian groups such that their enhanced power graphs are dominatable. In this paper, we try to answer their question. We prove that the enhanced power graph of finite group [Formula: see text] is dominatable if and only if there is a prime number [Formula: see text] such that [Formula: see text] and the Sylow [Formula: see text]-subgroups of [Formula: see text] are isomorphic to either a cyclic group or a generalized quaternion group.
The Fibonacci cube Γ n is obtained from the n-cube Q n by removing all the vertices that contain two consecutive 1s. It is proved that Γ n admits a perfect code if and only if n ≤ 3.
Let G be a finite group. The prime graph of G is denoted by Γ(G). Let G be a finite group such that Γ(G) = Γ(Dn(5)), where n ≥ 6. In the paper, as the main result, we show that if n is odd, then G is recognizable by the prime graph and if n is even, then G is quasirecognizable by the prime graph. Нехай G-скiнченна група. Простий граф групи G позначимо через Γ(G). Нехай G-скiнченна група така, що Γ(G) = Γ(Dn(5)), де n ≥ 6. Як основний результат роботи доведено, що для непарних n група G розпiзнається простим графом, а для парних n група G є такою, що квазiрозпiзнається простим графом.
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