WIENER AND HARARY INDICES OF THE GENERALIZED ZERO-DIVISOR GRAPH

Anil Khairnar; Anita Lande; Geetali Bhavar; Nanasaheb Phatangare

WIENER AND HARARY INDICES OF THE GENERALIZED ZERO-DIVISOR GRAPH

Abstract

Let $R$ be a commutative ring, and let $Z^*(R)$ denote the set of all non-zero zero-divisors in $R$. The generalized zero-divisor graph of $R$, denoted by $\Gamma'(R)$, is a simple (undirected) graph with the vertex set $Z^*(R)$, and two distinct vertices $x$ and $y$ are adjacent in $\Gamma'(R)$ if and only if $x^ny=0$ or $y^nx=0$, for some positive integer $n$. The Wiener and Harary indices are topological indices widely used in chemistry to analyze the structures of molecules. In this paper, we determine the Wiener and Harary indices of $\Gamma'(\mathbb Z_{n})$ for $n=pq,p^2q,p^2q^2$, where $p, q$ are distinct primes.

Keywords

Thanks

The authors are deeply grateful to the anonymous Reviewers for all valuable comments and suggestions, which helped us to improve the quality of the manuscript.

References

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Details

Primary Language

English

Subjects

Algebra and Number Theory, Combinatorics and Discrete Mathematics (Excl. Physical Combinatorics)

Journal Section

Research Article

Authors

Anil Khairnar This is me
0000-0003-2187-6362
India

Anita Lande ^*
0009-0005-8542-6290
India

Geetali Bhavar This is me
0009-0005-2794-3046
India

Nanasaheb Phatangare This is me
0000-0003-3826-7037
India

Publication Date

October 6, 2026

Submission Date

July 9, 2025

Acceptance Date

October 16, 2025

Published in Issue

Year 2026 Volume: 16 Number: 10

IZ

https://izlik.org/JA24YX59AF

Cite

RIS / Bibtex

APA

Khairnar, A., Lande, A., Bhavar, G., & Phatangare, N. (2026). WIENER AND HARARY INDICES OF THE GENERALIZED ZERO-DIVISOR GRAPH. TWMS Journal of Applied and Engineering Mathematics, 16(10), 1273-1285. https://izlik.org/JA24YX59AF

AMA

1.Khairnar A, Lande A, Bhavar G, Phatangare N. WIENER AND HARARY INDICES OF THE GENERALIZED ZERO-DIVISOR GRAPH. JAEM. 2026;16(10):1273-1285. https://izlik.org/JA24YX59AF

Chicago

Khairnar, Anil, Anita Lande, Geetali Bhavar, and Nanasaheb Phatangare. 2026. “WIENER AND HARARY INDICES OF THE GENERALIZED ZERO-DIVISOR GRAPH”. TWMS Journal of Applied and Engineering Mathematics 16 (10): 1273-85. https://izlik.org/JA24YX59AF.

EndNote

Khairnar A, Lande A, Bhavar G, Phatangare N (October 1, 2026) WIENER AND HARARY INDICES OF THE GENERALIZED ZERO-DIVISOR GRAPH. TWMS Journal of Applied and Engineering Mathematics 16 10 1273–1285.

IEEE

[1]A. Khairnar, A. Lande, G. Bhavar, and N. Phatangare, “WIENER AND HARARY INDICES OF THE GENERALIZED ZERO-DIVISOR GRAPH”, JAEM, vol. 16, no. 10, pp. 1273–1285, Oct. 2026, [Online]. Available: https://izlik.org/JA24YX59AF

ISNAD

Khairnar, Anil - Lande, Anita - Bhavar, Geetali - Phatangare, Nanasaheb. “WIENER AND HARARY INDICES OF THE GENERALIZED ZERO-DIVISOR GRAPH”. TWMS Journal of Applied and Engineering Mathematics 16/10 (October 1, 2026): 1273-1285. https://izlik.org/JA24YX59AF.

JAMA

1.Khairnar A, Lande A, Bhavar G, Phatangare N. WIENER AND HARARY INDICES OF THE GENERALIZED ZERO-DIVISOR GRAPH. JAEM. 2026;16:1273–1285.

MLA

Khairnar, Anil, et al. “WIENER AND HARARY INDICES OF THE GENERALIZED ZERO-DIVISOR GRAPH”. TWMS Journal of Applied and Engineering Mathematics, vol. 16, no. 10, Oct. 2026, pp. 1273-85, https://izlik.org/JA24YX59AF.

Vancouver

1.Anil Khairnar, Anita Lande, Geetali Bhavar, Nanasaheb Phatangare. WIENER AND HARARY INDICES OF THE GENERALIZED ZERO-DIVISOR GRAPH. JAEM [Internet]. 2026 Oct. 1;16(10):1273-85. Available from: https://izlik.org/JA24YX59AF