  In this paper, we study Grover's search algorithm focusing on continuous-time
quantum walk on graphs. We propose an alternative optimization approach to
Grover's algorithm on graphs that can be summarized as follows: instead of
finding specific graph topologies convenient for the related quantum walk, we
fix the graph topology and vary the underlying graph Laplacians. As a result,
we search for the most appropriate analytical structure on graphs endowed with
fixed topologies yielding better search outcomes. We discuss strategies to
investigate the optimality of Grover's algorithm and provide an example with an
easy tunable graph Laplacian to investigate our ideas.
