Spectrum and Energy of the Mobius Function Graph of Finite Cyclic Groups

The graph’s difference between the degree and the adjacency matrices is represented by the Laplacian matrix. The Mobius function graph \(M(\textrm{G})\) of a finite group \(\textrm{G}\) , has same vertex set of \(\textrm{G}\) and vertices \(\mathfrak {a}\) and \(\mathfrak {b}\) being adjacent if and only if \( \mu ( \mid \mathfrak {a} \mid \mid \mathfrak {b}\mid ) \) = \(\mu ( \mid \mathfrak {a} \mid ) \mu ( \mid \mathfrak {b} \mid )\) . In this paper, we identify the adjacency and the Laplacian spectrum, energy, and the Laplacian energy of the Mobius function graph of finite cyclic groups.