The Sierpiński product of graphs

Kovič, Jurij (author), Pisanski, Tomaž (author), Zemljič, Sara Sabrina (author), Žitnik, Arjana (author)

In this paper we introduce a product-like operation that generalizes the construction of the generalized Sierpiński graphs. Let $G, \, H$ be graphs and let $f: V(G) \to V(H)$ be a function. Then the Sierpiński product of graphs $G$ and $H$ with respect to $f$, denoted by $G\otimes_f H$, is defined as the graph on the vertex set $V(G) \times V(H)$, consisting of $|V(G)|$ copies of $H$; for every edge $\{g, g'\}$ of $G$ there is an edge between copies $gH$ and $g'H$ of form $\{(g, f(g'), (g', f(g))\}$. Some basic properties of the Sierpiński product are presented. In particular, we show that the graph $G\otimes_f H$ is connected if and only if both graphs $G$ and $H$ are connected and we present some conditions that $G, \, H$ must fulfill for $G\otimes_f H$ to be planar. As for symmetry properties, we show which automorphisms of $G$ and $H$ extend to automorphisms of $G\otimes_f H$. In several cases we can also describe the whole automorphism group of the graph $G\otimes_f H$. Finally, we show how to extend the Sierpiński product to multiple factors in a natural way. By applying this operation $n$ times to the same graph we obtain an alternative approach to the well-known $n$-th generalized Sierpiński graph.

• Language: English
• Keywords: Sierpiński graphs, graph products, connectivity, planarity, symmetry
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Publication date: 01.01.2023
• Year: 2023
• Number of pages: art. P1.01 (25 str.)
• Numbering: Vol. 23, no. 1
• PID: 20.500.12556/RUL-155088
• UDC: 519.17
• ISSN on article: 1855-3966
• DOI: 10.26493/1855-3974.1970.29e
• COBISS.SI-ID: 143319043
• Publication date in RUL: 19.03.2024

Record is a part of a journal

• Title: Ars mathematica contemporanea
• Publisher: Društvo matematikov, fizikov in astronomov, Društvo matematikov, fizikov in astronomov, Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije
• ISSN: 1855-3966
• COBISS.SI-ID: 239049984

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Secondary language

• Language: Slovenian
• Keywords: grafi Sierpińskega, produkti grafov, povezanost, ravninskost, simetrija