Spectra of Manhattan Products of Directed Paths Pn#P2

概要

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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.