Graph of Orthogonal Idempotent Elements in Rings of Integers Modulo n With Spectral Clustering

The zero‐divisor graph provides a relationship between abstract algebra and graph theory. Based on this concept, a new graph is defined using the orthogonal idempotent elements in a ring. The vertices of this graph are the set of idempotent elements, with edges connecting pairs of orthogonal idempotent elements. The main result presented in the paper is the construction of these new graphs by studying some of their properties, such as the Hosoya polynomial, for any ring of integers modulo n .