Laplacian spectrum of the unit graph associated to the ring of integers modulo pq

Abstract

Let $ R $ be a ring and $ U(R) $ be the set of unit elements of $ R $. The unit graph $ G(R) $ of $ R $ is the graph whose vertices are all the elements of $ R $, defining distinct vertices $ x $ and $ y $ to be adjacent if and only if $ x + y \in U(R) $. The Laplacian spectrum of $ G(\mathbb{Z}_n) $ was studied when $ n = p^{m} $, where $ p $ is a prime and $ m $ is a positive integer. Consequently, in this paper, we study the Laplacian spectrum of $ G(\mathbb{Z}_n) $, for $ n = p_1p_2...p_k $, where $ p_i $ are distinct primes and $ i = 1, 2, ..., k $.

Keywords

• unit graph
• laplacian matrix
• laplacian spectrum
• direct product
• ring of integers modulo n