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 adjacent or incident. In this paper, we investigate 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 $r>2$ and $r+1$ is not a Laplacian eigenvalue of $G$, then $\mathcal{T}(G)$ fails to exhibit Laplacian perfect pair state transfer. We also prove that if $G$ is a complete graph on more than three vertices, then $\mathcal{T}(G)$ fails to exhibit Laplacian perfect pair state transfer. Further, we prove that under some mild conditions, $\mathcal{T}(G)$ exhibits Laplacian pretty good pair state transfer, where $G$ is an $r$-regular graph such that $r>2$ and $r+1$ is not a Laplacian eigenvalue of $G$. We use these conditions to obtain infinitely many total graphs exhibiting Laplacian pretty good pair state transfer.

💡 Research Summary

The paper investigates quantum state transfer on the total graph 𝒯(G) of a regular graph G, focusing on Laplacian perfect pair state transfer (LP‑PST) and Laplacian pretty‑good pair state transfer (LP‑PGST). A pair state is represented by the vector e_a − e_b, and the transition matrix is U_G(t)=exp(−i t L_G), where L_G is the Laplacian of G. LP‑PST occurs if there exists a non‑zero time t₀ such that the inner product |(e_a − e_b)^T U_G(t₀)(e_c − e_d)| equals 1; LP‑PGST requires that for every ε>0 a time t can be found with the inner product within ε of 1.

The authors first recall the Chen‑Godsil criteria for LP‑PST: (i) the two pair states must be Laplacian strongly cospectral, (ii) the eigenvalues involved must be either all integers or all quadratic integers of the form ½(x + y√Δ) with Δ square‑free, and (iii) a parity condition linking the eigenvalue differences and √Δ must hold. These conditions are necessary and sufficient.

Using the known spectral decomposition of the total graph (Theorem 2.5), the Laplacian eigenvalues of 𝒯(G) are expressed in terms of the eigenvalues θ_j of G as θ_j^± = r + 2 + 2θ_j ± √{(r + 2)² − 4θ_j²}, where r is the regular degree of G. Additional eigenvalues 2r + 2 (or 3r in the bipartite case) also appear.

In Section 3 the paper proves that LP‑PST never occurs on 𝒯(G) under fairly mild hypotheses. Lemma 3.1 (non‑bipartite G) and Lemma 3.2 (bipartite G) show a symmetry: θ_j^+ belongs to the support Φ_ab of a pair state if and only if θ_j^− does. Combining this with the strong‑cospectral requirement forces the support to be empty or to contain eigenvalues that cannot satisfy the integer or quadratic‑integer condition of Chen‑Godsil. Consequently, for any r‑regular G with r>2 and with r + 1 not a Laplacian eigenvalue of G, 𝒯(G) cannot exhibit LP‑PST (Theorem 3.4‑3.6). The authors also treat the special case of complete graphs K_n (n>3) directly, showing that their total graphs also lack LP‑PST.

Section 4 turns to LP‑PGST. The key tool is Kronecker’s simultaneous approximation theorem (Theorem 2.3), which guarantees that linear combinations of irrational numbers can be approximated arbitrarily closely by rational numbers. Lemma 2.4 ensures that the set {1, √Δ : Δ square‑free > 1} is linearly independent over ℚ. The authors identify conditions under which the eigenvalue differences of 𝒯(G) contain at least one irrational component (coming from √Δ). When r>2 and r + 1 is not a Laplacian eigenvalue of G, and G possesses an eigenvalue θ_j such that (r + 2)² − 4θ_j² is not a perfect square, the corresponding θ_j^± are of the quadratic‑integer type with a non‑trivial √Δ. By selecting a pair of vertices (or edges) that are strongly cospectral (which is guaranteed by Lemma 3.1/3.2), the authors apply Kronecker’s theorem to construct times t_k at which the transition amplitude approaches 1, thereby establishing LP‑PGST for 𝒯(G) (Theorem 4.5).

Finally, the paper provides explicit infinite families of regular graphs satisfying the required spectral conditions. Examples include certain circulant graphs C_n with n≡1 (mod 4), regular bipartite graphs derived from hypercubes, and families of line graphs of complete graphs. For each family, r>2 and r + 1 is not a Laplacian eigenvalue, while the discriminant Δ is square‑free, guaranteeing the existence of LP‑PGST on their total graphs.

Overall, the work extends the theory of Laplacian‑based quantum state transfer to the total graph construction, delivering a clear dichotomy: perfect pair transfer is essentially impossible for regular total graphs beyond trivial cases, whereas pretty‑good pair transfer occurs abundantly under natural spectral constraints. The results enrich the understanding of how graph operations affect quantum transport properties and open avenues for designing networks that support high‑fidelity, albeit approximate, quantum communication.