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arXiv · 2604.26603

On main eigenvalues of zero-divisor graphs of reduced rings

Sakshi Jain·Y. M. Borse·R. Barabde
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Abstract

The problem of characterizing graphs with a prescribed number of main eigenvalues is a long-standing problem in spectral graph theory. Although some constructions are known, only a few produce infinite families of simple connected graphs with exactly $s \ge 2$ main eigenvalues. Zero-divisor graphs form a well-structured class of algebraic graphs whose spectra can be described explicitly using equitable partitions, making them a convenient setting to study main eigenvalues. In this paper, we prove that the zero-divisor graphs of reduced rings provide an infinite family of simple connected graphs with exactly $s$ main eigenvalues, and that certain induced bipartite subgraphs also have exactly $s$ main eigenvalues for any positive integer $s$.

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Sakshi Jain, Y. M. Borse, R. Barabde. 2026-05-08. On main eigenvalues of zero-divisor graphs of reduced rings. https://arxiv.org/abs/2604.26603

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