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All Journal CAUCHY: Jurnal Matematika Murni dan Aplikasi Syntax Literate: Jurnal Ilmiah Indonesia
Musyarrofah, Ayunda Faizatul
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Humanities Education Environmental Science Law, Crime, Criminology & Criminal Justice Mathematics Social Sciences Other

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Zero Divisor Graph of Quotient Ring Musyarrofah, Ayunda Faizatul; Krisnawati, Vira Hari; Hidayat, Noor
CAUCHY: Jurnal Matematika Murni dan Aplikasi Vol 9, No 2 (2024): CAUCHY: JURNAL MATEMATIKA MURNI DAN APLIKASI
Publisher : Mathematics Department, Universitas Islam Negeri Maulana Malik Ibrahim Malang

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.18860/ca.v9i2.29008

Abstract

Recently, a lot of research has been carried out regarding graphs built from algebraic structures, including ring structures. One important example of a graph constructed from a ring is the zero divisor graph. For a commutative ring R, the zero divisor graph Γ(R) is defined as a simple graph with vertices that are non-zero zero divisors of R, and two distinct vertices are adjacent if and only if the product of the vertices is equal to zero. In this paper, we investigate the zero divisor graph of the quotient ring ℤp[x]/⟨x5⟩ with prime p. More precisely, we characterize some graph properties, including the order, size, adjacency matrix, degree, distance, diameter, girth, clique number, and chromatic number of Γ(ℤp[x]/⟨x5⟩).
Indeks Wiener dari Graf Identitas dan Graf Pangkat pada Grup Siklis Berhingga Darmajid, Darmajid; Hidayat, Noor; Wicaksono, Wildan Bagus; Musyarrofah, Ayunda Faizatul
Syntax Literate Jurnal Ilmiah Indonesia
Publisher : Syntax Corporation

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.36418/syntax-literate.v9i7.16994

Abstract

Graf identitas atas suatu grup didefinisikan sebagai graf dengan himpunan titiknya berupa unsur- unsur grup dan dua titik dihubungkan oleh sebuah sisi jika hasil kalinya merupakan unsur identitas atau tepat salah satu dari kedua titik merupakan unsur identitas pada grup. Graf pangkat atas suatu grup didefinisikan sebagai graf dengan himpunan titiknya berupa unsur-unsur grup dan dua titik dihubungkan oleh sebuah sisi jika satu titik dapat dituliskan sebagai perpangkatan dari titik lainnya. Pada penelitian ini dikaji formulasi indeks Wiener dari graf identitas dan graf pangkat pada grup siklik berhingga. Hasil formulasi indeks Wiener dari graf identitas terbagi atas grup siklis orde 1, orde ganjil lebih dari 1 dan orde genap sedangkan dari graf pangkat difokuskan pada grup siklis berode perpangkatan bilangan prima dan perkalian dua prima berbeda.
Some Properties of Prime Graph of Cartesian Product of the Ring Krisnawati, Vira Hari; Musyarrofah, Ayunda Faizatul; Hidayat, Noor; Fatimah, Farah Maulidya
CAUCHY: Jurnal Matematika Murni dan Aplikasi Vol 11, No 1 (2026): CAUCHY: JURNAL MATEMATIKA MURNI DAN APLIKASI
Publisher : Mathematics Department, Universitas Islam Negeri Maulana Malik Ibrahim Malang

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.18860/cauchy.v11i1.32154

Abstract

The prime graph of the ring R, (PG(R)) is a graph which set of vertices consists of elements of R and two different vertices are adjacent if their product in the ring is zero. We study the prime graph of cartesian product of the ring Z_(p_1 )×Z_(p_2 ) for distinct prime numbers p_1 and p_2. We find that some properties of PG(Z_(p_1 )×Z_(p_2 ) ) such as order, size, the number of triangles, and Wiener. Further, we construct the line graph of PG(Z_(p_1 )×Z_(p_2 ) ) and calculate the order, size, and Wiener index of L(PG(Z_(p_1 )×Z_(p_2 ) )).
Co-Authors Darmajid, Darmajid Fatimah, Farah Maulidya Krisnawati, Vira Hari Noor Hidayat, Noor Wicaksono, Wildan Bagus