Laplacian Pair State Transfer on Edge Coronas

Abstract

Let $G$ be a graph with edge set $E(G)=\{e_1,\dots ,e_m\}$, and let $\mathcal{H}=(H_1,\dots ,H_m)$ be an $m$-tuple of graphs. The edge corona $G\diamond \mathcal{H}$ is obtained from the disjoint union of $G,H_1,\dots ,H_m$ by joining both end vertices of each edge $e_j$ to every vertex of $H_j$ for $1\le j\le m$. In this paper, we investigate Laplacian perfect pair state transfer and Laplacian pretty good pair state transfer on edge coronas. We establish necessary and sufficient conditions for the existence of Laplacian perfect pair state transfer. We also derive sufficient conditions for the non-existence of Laplacian perfect pair state transfer. Further, we identify conditions under which the absence of Laplacian perfect pair state transfer guarantees the existence of Laplacian pretty good pair state transfer. Additionally, we develop constructions based on Cartesian product and tensor product to obtain a few more infinite families of edge coronas admitting Laplacian pretty good pair state transfer. Finally, we investigate the existence of Laplacian pretty good pair state transfer on the edge corona of complete graphs.

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