The Manhattan product of directed paths
Pn and
Pm 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
Pn and
Pm is unique up to isomorphisms, which is called standard and denoted by
Pn#
Pm. 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
Pn#
′Pm. The characteristic polynomials of
Pn#
P2 and
Pn#
′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 (
P2n-1#
P2)=ev (
P2n#
P2) and ev (
P2n#
′P2)=ev (
P2n+2#
P2). The limit of the spectral distribution of
Pn#
P2 as n→∞ exists in the sense of weak convergence and its concrete form is obtained.
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