Laplacian Pair State Transfer on Total Graphs

The total graph of a graph $G$, denoted $\mathcal{T}(G)$, is defined as the graph whose vertex set is the union of the vertex set of $G$ and the edge set of $G$, such that two vertices of $\mathcal{T}(G)$ are adjacent if the corresponding elements of $G$ are either adjacent or incident. In this paper, we investigate the existence of Laplacian perfect pair state transfer and Laplacian pretty good pair state transfer on $\mathcal{T}(G)$, where $G$ is an $r$-regular graph. We prove that if $G$ is Laplacian integral, $r \geq 3$, and $r+1$ is not a Laplacian eigenvalue of $G$, then $\mathcal{T}(G)$ does not exhibit Laplacian perfect pair state transfer. In addition, we prove that under some mild conditions, $\mathcal{T}(G)$ exhibits Laplacian pretty good pair state transfer, where $r \geq 3$ and $r+1$ is not a Laplacian eigenvalue of $G$. Using these conditions, we obtain several infinite families of total graphs exhibiting Laplacian pretty good pair state transfer that fail to exhibit Laplacian perfect pair state transfer. We also prove that the total graph of the complete graph $K_n$ exhibits Pair-LPGST if and only if $n=3$.