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10 pages, 15 figures.-- PACS nrs.: 73.22.-f, 75.75.+a, 72.80.Rj, 85.75.-d. ; Density-functional calculations predict half-metallicity in zigzag single-walled carbon nanotubes of finite length with the two ends saturated with hydrogen. We have analyzed the change of the α- and β-spin electronic gaps under the influence of an electric field applied along the nanotube axis. The half-metallic behavior, in which the electronic gap is zero for one spin flavor and nonzero for the other, is obtained for a critical electric field of 3.0/w V/Å, where w is the length of the nanotube. This critical field is the same as that predicted for graphene nanoribbons. By a detailed analysis of the spin structure of the ground state, we show the relevance of the edge states, electronic states spatially localized at the carbon atoms of the nanotube boundaries, on the on-set of half-metallicity, and on the magnetic properties of the finite semiconducting zigzag nanotubes. ; Research supported by MEC of Spain (Projects No. MAT2005-06544-C03-01, No. MAT2005-06544-C03-03, No. FIS2004-06490-C03-00, and MONACEM), by Junta de Castilla y León (Project No. VA039A05), by Basque Government (Project No. IE05-151) under the ETORTEK Program (NANOMAT), and by the European Network of Excellence NANOQUANTA (Grant No. NM4-CT-2004-500198). ; Peer reviewed

We study the Laplacian spectrum of token graphs, also called symmetric powers of graphs. The k-token graph Fk(G) of a graph G is the graph whose vertices are the k-subsets of vertices from G, two of which being adjacent whenever their symmetric difference is a pair of adjacent vertices in G. In this paper, we give a relationship between the Laplacian spectra of any two token graphs of a given graph. In particular, we show that, for any integers h and k such that 1 ≤ h ≤ k ≤ n 2 , the Laplacian spectrum of Fh(G) is contained in the Laplacian spectrum of Fk(G). We also show that the doubled odd graphs and doubled Johnson graphs can be obtained as token graphs of the complete graph Kn and the star Sn = K1,n−1, respectively. Besides, we obtain a relationship between the spectra of the k-token graph of G and the k-token graph of its complement G. This generalizes to tokens graphs a wellknown property stating that the Laplacian eigenvalues of G are closely related to the Laplacian eigenvalues of G. Finally, the doubled odd graphs and doubled Johnson graphs provide two infinite families, together with some others, in which the algebraic connectivities of the original graph and its token graph coincide. Moreover, we conjecture that this is the case for any graph G and its token graph. ; This research of C. Dalfó and M.A. Fiol has been partially supported by AGAUR from the Catalan Government under project 017SGR1087 and by MICINN from the Spanish Government under project PGC2018-095471-B-I00. The research of C. Dalfó has also been supported by MICINN from the Spanish Government under project MTM2017-83271-R. The research of C. Huemer was supported by MICINN from the Spanish Government under project PID2019-104129GB-I00/AEI/10.13039/501100011033 and AGAUR from the Catalan Government under project 017SGR1336. F.J. Zaragoza Martínez acknowledges the support of the National Council of Science and Technology (Conacyt) and its National System of Researchers (SNI). This research has also received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 734922