Tree-Line graphs and their quantum walks

For a simple graph $\mathrm{\Gamma }$, a (bipartite)tree-line graph and a tree-graph of $\mathrm{\Gamma }$ can be defined. With a (bipartite)tree-line graph constructed by the function $(b)\mathrm{\ell }$, we study the continuous quantum walk on $(b){\mathrm{\ell }}^n\mathrm{\Gamma }$. 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.