Sains Malaysiana 50(2)(2021): 549-557

http://dx.doi.org/10.17576/jsm-2021-5002-25

On Diameter of Subgraphs of Commuting Graph in Symplectic Group for Elements of Order Three

(Diameter Subgraf bagi Graf Kalis Tukar Tertib dalam Kumpulan Simplektik bagi Unsur Berperingkat Tiga)

SUZILA MOHD KASIM¹ & ATHIRAH NAWAWI^1,2*

¹Institute for Mathematical Researc, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor Darul Ehsan, Malaysia

²Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor Darul Ehsan, Malaysia

Received: 1 April 2020/Accepted: 4 August 2020

ABSTRACT

Suppose  be a finite group and  be a subset of . The commuting graph, denoted by , is a simple undirected graph, where  being the set of vertex and two distinct vertices  are joined by an edge if and only if . The aim of this paper was to describe the structure of disconnected commuting graph by considering a symplectic group and a conjugacy class of elements of order three. The main work was to discover the disc structure and the diameter of the subgraph as well as the suborbits of symplectic groups ,  and . Additionally, two mathematical formulas are derived and proved, one gives the number of subgraphs based on the size of each subgraph and the size of the conjugacy class, whilst the other one gives the size of disc relying on the number and size of suborbits in each disc.

Keywords: Commuting graph; conjugacy class; disconnected graph; symplectic group

ABSTRAK

Andaikan  adalah satu kumpulan terhingga dan  adalah satu subset bagi . Graf kalis tukar tertib, ditatatandakan dengan  adalah graf mudah tidak terarah, yang  menjadi set bucu dan dua bucu berbeza  disambungkan oleh satu garisbucu jika dan hanya jika . Tujuan makalah ini adalah untuk memperincikan struktur graf kalis tukar tertib tidak berkait dengan mempertimbangkan kumpulan simplektik dan kelas konjugasi dengan unsur berperingkat tiga. Kerja utama adalah untuk memperoleh struktur cakera dan diameter subgraf tersebut juga suborbit bagi kumpulan simplektik ,  dan . Di samping itu, dua formula matematik diterbitkan dan dibuktikan, satu daripadanya memberikan bilangan subgraf berdasarkan kepada saiz setiap subgraf dan saiz kelas konjugasi, manakala yang satu lagi memberikan saiz cakera bergantung pada bilangan dan saiz suborbit dalam setiap cakera.

Kata kunci: Graf kalis tukar tertib; graf tidak berkait; kelas konjugasi; kumpulan simplektik

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*Corresponding author; email: athirah@upm.edu.my