On main eigenvalues of zero-divisor graphs of reduced rings
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| Format: | Preprint |
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2026
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| _version_ | 1866914541900660736 |
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| author | Jain, Sakshi Borse, Y. M. Barabde, R. |
| author_facet | Jain, Sakshi Borse, Y. M. Barabde, R. |
| contents | 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 $s$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_26603 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On main eigenvalues of zero-divisor graphs of reduced rings Jain, Sakshi Borse, Y. M. Barabde, R. Combinatorics Commutative Algebra 05C50, 11B39, 11C20 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 $s$. |
| title | On main eigenvalues of zero-divisor graphs of reduced rings |
| topic | Combinatorics Commutative Algebra 05C50, 11B39, 11C20 |
| url | https://arxiv.org/abs/2604.26603 |