On set difference of maximal cyclic subgroups of a pp-group

Let $G$ be a finite group. For maximal cyclic subgroups $M, N$ of $G$, we denote by $\text{d}(M,N)$ the minimum of number of elements in the set differences $M {\setminus} N$ and $N {\setminus} M$. The difference number $\delta(G)$ of $G$ is defined as the maximum of $\text{d}(M,N)$ as $M$ and $N$ vary over every pair of maximal cyclic subgroups of $G$. Whereas, the power graph $\Gamma_G$ of $G$ is the undirected simple graph with vertex set $G$ and two distinct vertices are adjacent if one of them is a positive power of the other. A connected graph $\Gamma$ is said to be cyclically separable if it has a vertex set whose deletion results in a disconnected subgraph with at least two components containing cycles. In this paper, we derive a relationship between the difference number and the power graph of a group. We prove that for a finite $p$-group $G$, $\delta(G) \geq 3$ if and only if $\Gamma_G$ is cyclically separable

Matching in power graphs of finite groups

Funding: The author Swathi V V acknowledges the support of Council of Scientific and Industrial Research, India (CSIR) (Grant No-09/874(0029)/2018-EMR-I), and DST, Government of India, ‘FIST’ (No.SR/FST /MS-I/2019/40).The power graph P(G) of a finite group G is the undirected simple graph with vertex set G, where two elements are adjacent if one is a power of the other. In this paper, the matching numbers of power graphs of finite groups are investigated. We give upper and lower bounds, and conditions for the power graph of a group to possess a perfect matching. We give a formula for the matching number for any finite nilpotent group. In addition, using some elementary number theory, we show that the matching number of the enhanced power graph Pe(G) of G (in which two elements are adjacent if both are powers of a common element) is equal to that of the power graph of G.Publisher PDFPeer reviewe