On Spectrum and Energy of Non-Commuting Graph for Group U_{6n}

Romdhini, Mamika Ujianita
Nawawi, Athirah
Al-Sharqi, Faisal
Al-Quran, Ashraf

Journal of the Indonesian Mathematical Society
Vol. 31 No. 4 (2025): DECEMBER

Let $G$ be a group and $Z(G)$ be the center of $G$. In this paper, we discuss a specific type of graph known as the non-commuting graph, denoted by $\Gamma_G$, whose vertex set contains all group elements excluding central elements, $G\backslash Z(G)$. This graph has to satisfy a condition in which $v_p,v_q \in G\backslash Z(G)$ where $v_p \neq v_q$, are adjacent whenever $v_p v_q\neq v_q v_p$. This paper presents the spectrum and energy of the non-commuting graph for $U_{6n}$, $\Gamma_{U_{6n}}$ associated with the adjacency, degree sum, and degree sum adjacency matrices and their energy relationship.