Laplacian state transfer in graphs with involutions

TL;DR

This paper studies perfect state transfer in graphs with involutions using generalized Laplacians, establishing conditions and constructions for PST in various graph classes, including paths, cycles, and bipartite graphs.

Contribution

It links PST in graphs with involutions to subgraph PST, and characterizes PST conditions for paths, cycles, and other graphs with added loops or edges.

Findings

Almost all unweighted planar graphs with two loops exhibit PST.

Paths on two vertices always admit PST; certain three-vertex paths do under specific conditions.

Adding edges or loops can enable PST in various graph classes.

Abstract

For $q\in\mathbb{R}\backslash\{0\}$, the generalized Laplacian of a graph $X$ is the matrix $\mathscr{L}=\Delta+qA$, where $\Delta$ is the degree matrix and $A$ is the adjacency matrix of $X$. In this paper, we investigate perfect state transfer (PST) on graphs with possible loops equipped with non-trivial involutions, where we take the generalized Laplacian matrix as the Hamiltonian of the underlying spin network. We establish an equivalence between the existence of PST between certain pair (or plus states) in such a graph and PST between vertices in a subgraph induced by the involution. This allows us to prove that for almost all simple unweighted planar graphs (resp., almost all simple unweighted trees), the assignment of loops of weight one to exactly two vertices in the graph produces PST between pair states relative to $\mathscr{L}$. We also show that a path on $n$ vertices admits PST between end vertices relative to $\mathscr{L}$ if and only if $n =2$, or $(n,q)=(3,\frac{k^2-l^2}{8l^2})$ where $k>l$ are integers with $k \not\equiv l \pmod{2}$. For cycles, we show that the addition of an extra edge does not yield PST between vertices relative to Laplacian and signless Laplacian matrices. Furthermore, we show that the addition of a few suitable edges (including loops) in complete bipartite graphs, cycles, and paths yields PST between pair states.