Malaysian Journal of Mathematical Sciences, December 2025, Vol. 19, No. 4
On Laplacian Commutativity of Graphs
Nithya, S. and Kuttipulackal, P.
Corresponding Email: nithyas@uoc.ac.in
Received date: 6 February 2025
Accepted date: 28 May 2025
Abstract:
This paper introduces the notion of Laplacian commutativity of graphs among well known classes of graphs. Two graphs are Laplacian commutative if their Laplacian matrices commute. The commutativity of the Laplacian matrix of a graph $G$ with its complement, $G'$, and its $k-$complement, ${G_k}^{\mathscr{P}}$ is also examined. Laplacian commutativity depends on the partition $ \mathscr{P}$ of vertex set of $G$, $V_G $ and ${G_k}^{\mathscr{P}}$. Some necessary and sufficient conditions on the partition $ \mathscr{P} $ are described for the Laplacian commutativity of cycle $ C_{n} $ with ${(C_{n})_k^{\mathscr{P}}}$.
Keywords: Laplacian matrix; adjacency matrix; commutativity; graph complement; $k-$complement; $GH-$path
