Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 19 (2024), 109 -- 126

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This work is licensed under a Creative Commons Attribution 4.0 International License.

EXPLORING THE STRUCTURE AND PROPERTIES OF IDEAL-BASED ZERO-DIVISOR GRAPHS IN INVOLUTION NEAR RINGS

Masreshaw Walle Abate and Wang Yao

Abstract. For an involution near ring 𝒩 and its ideal ℐ, the text introduces an involution ideal-based zero- divisor graph Γℐ*(𝒩) which is an undirected graph with vertex set

{ x ∈ 𝒩 - ℐ: x𝒩y ⊂ ℐ (or y𝒩x ⊂ ℑ ) for some y ∈ 𝒩-ℐ},
where two distinct vertices x and y are adjacent if and only if y𝒩x* ⊂ ℐ or x𝒩y* ⊂ ℐ. The paper provides characterizations of Γℐ*(𝒩) when it forms a complete graph or a star graph. It also explores the structure of Γℐ*(𝒩), investigates its properties like connectedness with diam(Γℐ*(𝒩)) ≤ 3 and analyzes the connection of Γℐ*(𝒩) with Γℐ*(𝒩/ℐ). Furthermore, the paper discusses the chromatic number and clique number of the graph. It also characterizes all right permutable *-near-rings 𝒩 for which the graph Γℐ*(𝒩) can be with a finite chromatic number.

2020 Mathematics Subject Classification: 05C25, 16W10, 16Y30, 13A70
Keywords: near ring, zero-divisor-graph, ideal based zero-divisor graph, ideal of *-near-ring, graph coloring, chromatic number

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Masreshaw Walle Abate
School of Mathematics and Statistics,
Nanjing University of Information Science and Technology,
219 Ningliu Road, Nanjing 210044, China.
and
Department of Mathematics, Dilla University, 419 Dilla, Ethiopia
e-mail: masswalle@yahoo.com, 20205115001@nuist.edu.cn

Wang Yao (corresponding author)
School of Mathematics and Statistics,
Nanjing University of Information Science and Technology,
219 Ningliu Road, Nanjing 210044, China.
e-mail: wangyao@nuist.edu.cn

http://www.utgjiu.ro/math/sma