Signless Laplacian State Transfer on Vertex Complemented Coronae
Ke-Yu Zhu, Gui-Xian Tian, Shu-Yu Cui
TL;DR
This work develops a comprehensive framework for signless-Laplacian quantum state transfer on vertex complemented coronae. It derives explicit spectral decompositions for G~∘H when G and H are regular, enabling closed-form transition amplitudes and rigorous PST/PGST analysis (Theorems 3.1–3.2). Through periodicity criteria (Theorems 4.2–4.3) and targeted nonperiodicity conditions (Theorems 4.4–4.6 and Examples 1–2), the authors establish extensive PST obstructions while identifying PGST-rich constructions, notably via Theorem 5.1 and Theorem 5.2 that leverage irrationality/Diophantine-approximation arguments. The results extend signless-Laplacian state transfer literature to vertex complemented coronae, offering concrete guidelines for constructing graphs with PST or PGST and highlighting cases where PST is impossible but PGST remains achievable (e.g., Example 3).
Abstract
Given a graph $G$ with vertex set $V(G)=\{v_1,v_2,\ldots,v_{n_1}\}$ and a graph $H$ of order $n_2$, the vertex complemented corona, denoted by $G\tilde{\circ}{H}$, is the graph produced by copying $H$ $n_1$ times, with the $i$-th copy of $H$ corresponding to the vertex $v_i$, and then adding edges between any vertex in $V(G)\setminus\{v_{i}\}$ and any vertex of the $i$-th copy of $H$. The present article deals with quantum state transfer of vertex complemented coronae concerning signless Laplacian matrix. Our research investigates conditions in which signless Laplacian perfect state transfer exists or not on vertex complemented coronae. Additionally, we also provide some mild conditions for the class of graphs under consideration that allow signless Laplacian pretty good state transfer.