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标题：Spectra of Manhattan Products of Directed Paths Pn#P2
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作者：Nobuaki OBATA
期刊名称：Interdisciplinary Information Sciences
印刷版ISSN：1340-9050
电子版ISSN：1347-6157
出版年度：2012
卷号：18
期号：1
页码：43-54
DOI：10.4036/iis.2012.43
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