Asymptotic spectral distributions of Manhattan products of $C_{n}\sharp P_{m}$

Abstract

The Manhattan product of directed cycles $C_{n}$ and directed paths $P_{m}$ is a diagraph. Recently, in quantum probability theory, several authors have studied the spectrum of graph, as mentioned also by A. Hora and N. Obata. In the paper, we study asymptotic spectral distribution of the Manhattan products of simple digraphs-$C_{n}\sharp P_{m}$. The limit of the spectral distribution of $C_{n}\sharp P_{2}$ as $n\rightarrow \infty$ exists in the sense of weak convergence, and its concrete form is obtained. We insist on the fact that this note does not contain any new results, which is only some parallel results with Obata (Interdiscip Inf Sci 18(1):43–54, 2012) or Obata (Ann Funct Anal 3:136–144, 2012). But, we have only been written to convey the information from quantum probability to spectral analysis of graph.