\( n \)-th central graph of a group (Q2822177)

In this paper, the \( n \)-th central graph \(\Gamma_{z}^{n}(G)\) of a group with \( Z_{n}(G)\) being the \( n \)-th term of upper central series of \( G \) is studied for every \( n \geq 1 \). The authors investigate the structure of the graph \(\Gamma_{z}^{n}(G)\). Some numerical invariants -- dominating, chromatic and independence numbers -- are determined, as well as planarity and energy. The properties of the \( (r,s,t)\)-group associated with \(\Gamma_{z}^{n}(G)\) are investigated and conditions are obtained in order that two \( n \)-th central graphs are isomorphic. Moreover, the authors prove that the Cayley graph's properties are inherited to the \( n \)-th central graph.