Spectra of Manhattan Products of Directed Paths P_n#P₂

Abstract

The Manhattan product of directed paths P_n and P_m is a digraph, where the underlying graph is the n×m lattice and each edge is given direction in such a way that left and right directed horizontal lines are placed alternately, and so are up and down directed vertical lines. Unless both m and n are even, the Manhattan product of P_n and P_m is unique up to isomorphisms, which is called standard and denoted by P_n#P_m. If both m and n are even, there is a Manhattan product which is not isomorphic to the standard one. It is called non-standard and denoted by P_n#^′P_m. The characteristic polynomials of P_n#P₂ and P_n#^′P₂ are expressed in terms of the Chebychev polynomials of the second kind, and their spectra (eigenvalues with multiplicities) are thereby determined explicitly. In particular, it is shown that ev (P_2n-1#P₂)=ev (P_2n#P₂) and ev (P_2n#^′P₂)=ev (P_2n+2#P₂). The limit of the spectral distribution of P_n#P2 as n→∞ exists in the sense of weak convergence and its concrete form is obtained.