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Perfect non-commuting graphs of matrices over chains

Sarika Devhare

Center for Advanced Study, Department of Mathematics, Savitribai Phule Pune University, Pune 411007, India

E-mail Address: sarikadevhare@gmail.com

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and

Vinayak Joshi

http://orcid.org/0000-0001-9105-4634

Center for Advanced Study, Department of Mathematics, Savitribai Phule Pune University, Pune 411007, India

E-mail Address: vvjoshi@unipune.ac.in

E-mail Address: vinayakjoshi111@yahoo.com

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https://doi.org/10.1142/S1793830917500495Cited by:0 (Source: Crossref)

Abstract

In this paper, we study the non-commuting graph $Γ (M_{3}^{su} (C_{n}))$ of strictly upper triangular $3 \times 3$ matrices over an $n$ -element chain $C_{n}$ . We prove that $Γ (M_{3}^{su} (C_{n}))$ is a compact graph. From $Γ (M_{3}^{su} (C_{n}))$ , we construct a poset $P$ . We further prove that $P \oplus 1$ is a dismantlable lattice and its zero-divisor graph is isomorphic to $Γ (M_{3}^{su} (C_{n}))$ . Lastly, we prove that $Γ (M_{3}^{su} (C_{n}))$ is a perfect graph.

Keywords:

AMSC: Primary 05C15, Secondary 06A07

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