On the  Non-commuting Graph Associated to a Finite-dimensional Lie Algebra

Ahmad Erfanian; Akram Chareh Khah Moghaddam; Afsaneh  Shamsaki

doi:10.22342/jims.v32i1.1643

Ahmad Erfanian ⁽¹⁾ , Akram Chareh Khah Moghaddam ⁽²⁾ , Afsaneh Shamsaki ⁽³⁾

(1) Department of Pure Mathematics, Ferdowsi University of Mashhad, Iran, Islamic Republic of,

(2) Department of Pure Mathematics and Center of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad, Iran, Islamic Republic of,

(3) Department of Pure Mathematics, Ferdowsi University of Mashhad, Iran, Islamic Republic of

https://doi.org/10.22342/jims.v32i1.1643

Issue
Vol. 32 No. 1 (2026): MARCH

Submitted
January 28, 2024

Accepted
November 15, 2025

Published
April 17, 2026

Keywords:

Non-commuting graph, Lie algebra

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Abstract

In this paper, we define the non-commuting graph associated to a Lie algebra L and obtain some basic graph properties such as connectivity, diameter, girth, Hamiltonian and Eulerian. Moreover, planarity, outer planarity and isomorphism between two such graphs are also discussed in the paper.

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References

I. Beck, “Coloring of commutative rings,” Journal of algebra, vol. 116, no. 1, pp. 208–226, 1988. https://doi.org/10.1016/0021-8693(88)90202-5.

A. Abdollahi, S. Akbari, and H. Maimani, “Non-commuting graph of a group,” Journal of algebra, vol. 298, no. 2, pp. 468–492, 2006. https://doi.org/10.1016/j.jalgebra.2006.02.015.

B. Tolue and A. Erfanian, “Relative non-commuting graph of a finite group,” Journal of Algebra and its Applications, vol. 12, no. 02, p. 1250157, 2013. https://doi.org/10.1142/S0219498812501575.

J. A. Bondy, U. S. R. Murty, et al., Graph theory with applications, vol. 290. Macmillan London, 1976. https://www.iro.umontreal.ca/~hahn/IFT3545/GTWA.pdf.

R. Diestel, Graph Theory, 5th Ed., vol. 290. Springer, 2017.

K. Erdmann and M. J. Wildon, Introduction to Lie algebras, vol. 122. Springer, 2006. https://link.springer.com/book/10.1007/1-84628-490-2.

Authors

Ahmad Erfanian

Department of Pure Mathematics, Ferdowsi University of Mashhad

erfanian@um.ac.ir (Primary Contact)

Akram Chareh Khah Moghaddam

Department of Pure Mathematics and Center of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad

Afsaneh Shamsaki

Department of Pure Mathematics, Ferdowsi University of Mashhad

Author Biography

Ahmad Erfanian, Department of Pure Mathematics, Ferdowsi University of Mashhad

Department of Pure Mathematics

Erfanian, A., Moghaddam, A. C. K., & Shamsaki, A. (2026). On the Non-commuting Graph Associated to a Finite-dimensional Lie Algebra. Journal of the Indonesian Mathematical Society, 32(1), 1643. https://doi.org/10.22342/jims.v32i1.1643

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1. INTRODUCTION

Research topics in algebraic structures and graph theory have led to several exciting questions and results. Some papers have been written about assigning a graph to a group or ring and determining its algebraic properties. One can associate a graph to a groupGby various ways. For example, zero-divisor graph of a commutative ring, non-commuting graph of a group and relative non-commuting graph of a group are defined (see [1,2,3]). One of the associated graphs to a group \(G\)is the non-commuting graph, whose vertices are \(G \setminus Z ( G )\) and two distinct verticesxandyare adjacent if and only if \(x y \neq y x\) . Some results in group theory may be extended to Lie algebra. These conclusions are not always similar and even they are diferent in many situations. Recently, Lie algebras have attracted some authors due to wide applications in other sciences, especially physics. Understanding the structure of fundamental particles and the formulation of quantum theories in highenergy physics makes it inevitable for physicists to be familiar with Lie algebras.

In this paper, a graph \(G _ { L }\) is associated to a Lie algebra \(L\) and the properties of this kind of graph are investigated. Let \(L\) be a Lie algebra and \(Z ( L )\) be its center. We defined \(G _ { L }\) as follows: Take \(L \setminus Z ( L )\) the vertices of \(G _ { L }\) and join two distinct verticesxandywhenever \([ x , y ] \neq 0\) . We call \(G _ { L }\) the non-commuting graph of a Lie algebra L. All graphs are considered undirected and simple (with no loops or multiple edges). For every graph, we denote the set of vertices and the edges of \(G _ { L }\) by \(V ( G _ { L } )\) and \(E ( G _ { L } )\) , respectively. If \(L\) is abelian, then \(G _ { L }\) is null graph. Our main goal is to detect the properties of a Lie algebra \(L\) with its associated graph \(G _ { L }\) and vice verse.

In section 2, we study some properties of \(G _ { L }\) for a non-abelian Lie algebra. In addition, some results are obtained about connectivity, Hamiltonian, Eulerian properties, the girth and diameter of \(G _ { L } .\) . In section 3, we determine under what conditions of \(L\)c the graph \(G _ { L }\) is planar and outerplanar. In the last section, we consider isomorphism between two non-commuting graphs and show that under some conditions two Lie algebras have equal elements. In the rest of this section, we recall some concepts of graphs and Lie algebras that are used in the next sections.

Let \(\Gamma\) be an arbitrary graph. The degree of a vertex \(v \in V ( \Gamma )\) denoted by deg(v), is the number of edges, which are incident tov. The minimum and maximum degrees are be denoted by \(\delta(\Gamma)\) and \(\Delta ( \Gamma )\) , respectively. Also, \(\left|V(\Gamma)\right|\) is called the size of \(\Gamma\). If the degree of all vertices \(\Gamma\) is equal to \(| V ( \Gamma ) | - 1\) , then \(\Gamma\) is a complete graph. This graph with \(n\) vertices is denoted by \(K _ { n }\) . A graph \(\Gamma\) is regular if the degrees of all vertices of \(\Gamma\) are the same. A path of length n consists of vertices from \(v _ { 0 }\) to \(v _ { n }\) and a sequence of distinct \(e _ { i }\) such that \(e _ { i } = \{ v _ { i - 1 } , v _ { i } \}\) and \(1 \leq i \leq n\) A cycle is a path in a graph that starts and ends at the same vertex. A graph is connected when there is at least a path for every pair of vertices. Ifvandware two vertices of \(\Gamma\), then \(d ( v , w )\) is the length of the shortest path between v and w. The longest distance between all pairs of vertices is called the diameter of \(\Gamma\) and denoted by diam(\(\Gamma\)). The girth of \(\Gamma\), girth(\(\Gamma\)), is the length of the shortest cycle in graph \(\Gamma\). In graph theory, a planar graph is a graph that can be drawn such that none of the edges cross each other. The following theorem is an important tool in proving some theorems related to the planar graphs in section 3.

Theorem 1.1. [4, Corollary 9.5.3]If\(\Gamma\)is a simple and planar, then\(\delta ( \Gamma ) \leq 5\)

If a graph can be drawn in the plane without any edge crossings, such that all vertices lie on the boundary of the outer face is called an outerplanar graph. Every outerplanar graph is planar, but the converse of fact is not true. For example, the complete graph with four vertices is the smallest planar graph that is not outerplanar.

Theorem 1.2. [5, Chaper 10]Every subgraph of an outerplanar contains a vertex with degree at most 2.

An Eulerian cycle is cycle that visits every edge of a graph exactly once. If

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On the Non-commuting Graph Associated to a Finite-dimensional Lie Algebra

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Ahmad Erfanian, Department of Pure Mathematics, Ferdowsi University of Mashhad

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