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On gcd-graphs over finite rings
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abstract
Gcd-graphs represent an interesting and historically important class of integral graphs. Since the pioneering work of Klotz and Sander, numerous incarnations of these graphs have been explored in the literature. In this article, we define and establish some foundational properties of gcd-graphs defined over a general finite commutative ring. In particular, we investigate the connectivity and diameter of these graphs. Additionally, when the ring is a finite symmetric $\mathbb{Z}/n$-algebra, we give an explicit description of their spectrum using the theory of Ramanujan sums that gives a unified treatment of various results in the literature.
Forward citations
Cited by 2 Pith papers
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On divisibility relation graphs
D_n is planar exactly for n in {1, p, p^2, p^3, pq, p^2 q}, and its characteristic polynomial divides that of D_{n p q}, with a square divisibility when n has a simple prime factor.
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Supercharacters of finite abelian groups and applications to spectra of $U$-unitary Cayley graphs
A super-Cayley graph's spectrum is a super-Fourier transform of its connection set; for Frobenius rings this yields explicit spectral formulas and rationality criteria.
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