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All Journal CAUCHY: Jurnal Matematika Murni dan Aplikasi JTAM (Jurnal Teori dan Aplikasi Matematika)
Fatimah, Farah Maulidya
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Mathematics

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Prime Graph over Cartesian Product over Rings and Its Complement Fatimah, Farah Maulidya; Krisnawati, Vira Hari; Hidayat, Noor
JTAM (Jurnal Teori dan Aplikasi Matematika) Vol 7, No 3 (2023): July
Publisher : Universitas Muhammadiyah Mataram

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.31764/jtam.v7i3.14987

Abstract

Graph theory is a branch of algebra that is growing rapidly both in concept and application studies. This graph application can be used in chemistry, transportation, cryptographic problems, coding theory, design communication network, etc. There is currently a bridge between graphs and algebra, especially an algebraic structures, namely theory of graph algebra. One of researchs on graph algebra is a graph that formed by prime ring elements or called prime graph over ring R. The prime graph over commutative ring R (PG(R))) is a graph construction with set of vertices V(PG(R))=R and two vertices x and y are adjacent if satisfy xRy={0}, for x≠y. Girth is the shortest cycle length contains in PG(R) or can be written gr(PG(R)). Order in PG(R) denoted by |V(PG(R))| and size in PG(R) denoted by |E(PG(R))|. In this paper, we discussed prime graph over cartesian product over rings Z_m×Z_n and its complement. We focused only for m=p_1, n=p_2 and m=p_1, n=〖p_2〗^2, where p_1 and p_2 are prime numbers. Then, we obtained some properties related to order and size, degree, and girth. We also observe some examples. Moreover, we found that a correction in the statement of (Pawar & Joshi, 2019) about the complement graph over prime graph over a ring and gave a counter example for that. 
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 Krisnawati, Vira Hari Musyarrofah, Ayunda Faizatul Noor Hidayat, Noor