On the Adjacency Spectrum and Energy of the Nilpotent Graph of the Ring of Integers Modulo n

Abstract

Let ℤₙ be the ring of integers modulo n, with nil radical Nil(ℤₙ). Following Basnet, Sharma and Dutta, the nilpotent graph N(ℤₙ) has as its vertices the non-nilpotent elements of ℤₙ, and two distinct vertices x and y are joined precisely when is nilpotent. We prove that adjacency depends only on residues modulo the radical rad(n), and we use this to write the graph explicitly as a disjoint union of complete bipartite graphs, with a single complete graph appearing when rad(n) is even. The decomposition yields closed forms, in terms of the prime factorization of n alone, for the adjacency spectrum and the energy. The energy splits according to the parity of rad(n): it is
when rad(n) is odd and when rad(n) is even. We also give the number of connected components and the vertex degrees, and we check every formula on worked numerical examples.