Tree-Line graphs and their quantum walks

TL;DR

The paper develops a framework of tree-line graphs $\ell^n\Gamma$ and bipartite variants $b\ell^n\Gamma$ to study continuous quantum walks on derived graphs. It formalizes derived $k$-tree graphs via the tree map $T$ and establishes equitable partitions for $b\ell$-graphs, linking spectral properties to graph structure. Spectral analysis and examples show that periodic quantum walks depend on eigenvalue structure, with $C_n$ cycles generally non-periodic for $n>4$ and periodic behavior occurring in certain higher-order derived trees and quotient graphs. Appendices address interlacing limitations for multipartite graphs and foundational set-theoretic considerations, clarifying conditions under which key results hold and how they can be generalized or reinterpreted.

Abstract

For a simple graph $Γ$, a (bipartite)tree-line graph and a tree-graph of $Γ$ can be defined. With a (bipartite)tree-line graph constructed by the function $(b)\ell$, we study the continuous quantum walk on $(b)\ell ^n Γ$. An equitable partition of a bipartite tree-line graph is obtained by its corresponding derived tree graph. This paper also examines quantum walks on derived graphs, whose vertices represent their basis state.

Theorems & Definitions (21)

• Definition 1
• Definition 2: Tree-Line Graph constructing map $\ell$
• Definition 3: Bipartite Tree-Line Graph map $b\ell$
• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Definition 4