Presečni grafi komutativnih kolobarjev : delo diplomskega seminarja

Štebljaj, Živa (author), Oblak, Polona (mentor)

Presečni graf kolobarja je graf, v katerem so vozlišča pravi ideali v kolobarju, dve vozlišči pa sta sosedni, če imata ideala netrivialen presek. Obravnavamo presečne grafe komutativnih kolobarjev s posebnim poudarkom na kolobarjih ostankov pri deljenju $\mathbb{Z}_{n}$. Pokažemo, kdaj so presečni grafi komutativnih kolobarjev povezani, polni, brez triciklov, $r$-delni ali ravninski. Pri kolobarjih $\mathbb{Z}_{n}$ pokažemo tudi, kdaj so njihovi presečni grafi Eulerjevi ali Hamiltonovi.

• Language: Slovenian
• Keywords: presečni graf, kolobar, ideal, lokalni kolobar
• Work type: Final seminar paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2020
• PID: 20.500.12556/RUL-120195
• UDC: 512
• COBISS.SI-ID: 58096131
• Publication date in RUL: 17.09.2020

Secondary language

• Language: English
• Title: Intersection graphs of commutative rings

Abstract

An intersection graph of a ring is a graph in which the vertices are nontrivial ideals of the ring, and two vertices are adjacent if the ideals have a nontrivial intersection. We consider intersection graphs of commutative rings with special emphasis on the rings of integers modulo $n$. We discuss which commutative rings have graphs that are connected, complete, tricycle free, $r$-partite or planar. We also find rings $\mathbb{Z}_{n}$ that have Eulerian or Hamiltonian intersection graphs.

• Keywords: intersection graph, ring, ideal, local ring