Norio Konno
We consider an analogue of the well-known Riemann Hypothesis based on quantum walks on graphs with the help of the Konno-Sato theorem. Furthermore, we give some examples for complete, cycle, and star graphs.
This paper contains 5 sections, 4 theorems, 78 equations.
Theorem 1
Let $G$ be a simple connected graph with $V(G)= \{ v_1 , \ldots , v_n \}$ and $m$ edges. Then we have Here $\gamma$ is the Betti number of $G$ (i.e., $\gamma = m - n +1$ ), ${\bf I}_n$ is the $n \times n$ identity matrix, and ${\bf D}_n = [d_{ij}]$ is the $n \times n$ diagonal matrix with $d_{ii} = \deg v_i$ and $d_{ij} =0 \ (i \neq j)$.
