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All Journal CAUCHY: Jurnal Matematika Murni dan Aplikasi
Zyuhri, Syaifudin
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Mathematics

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The Triple Total Graph of The Ring Zn Kurniawan, Vika Yugi; Zyuhri, Syaifudin; Wiyono, Santoso Budi
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.v11i1.41488

Abstract

Let R be a commutative ring and let Z(R) denotes the set of zero-divisors of R. The triple total graph of R, denoted by TT(R), is a simple graph whose vertex set is R∖{0}. Two distinct vertices v1 and v2 are adjacent in TT(R) if and only if v1+v2∉Z(R) and there exists v3∈R∖{0}, with v3≠v1 and v3≠v2, such that v1+v3∉Z(R),  v2+v3∉Z(R), and v1+v2+v3∈Z(R).In this paper, we investigate the structural properties of the graph TT(Zn). We show that if n is even with n 2, then TT(Zn) is an empty graph. When n is prime with 2 n 11, the graph TT(Zn) is disconnected. In contrast, for prime integers n≥ 11, the graph becomes connected with diam(TT(Zn))=2 and gr(TT(Zn))=3. Moreover, each vertex has degree n−5, implying that the graph is (n−5)-regular and consequently both Eulerian and Hamiltonian. These results illustrate how the arithmetic nature of n determines the global structure of the triple total graph.
Co-Authors Santoso Budi Wiyono Vika Yugi Kurniawan