出版社：The Editorial Committee of the Interdisciplinary Information Sciences
摘要：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 # P 2 and P n #′ P 2 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 2 n -1# P 2)=ev ( P 2 n # P 2) and ev ( P 2 n #′ P 2)=ev ( P 2 n +2# P 2). The limit of the spectral distribution of P n # P 2 as n →∞ exists in the sense of weak convergence and its concrete form is obtained.
关键词：adjacency matrix;characteristic polynomial;Chebychev polynomials of the second kind;digraph;eigenvalue;Manhattan product;spectrum
