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Median Eigenvalues of Bipartite Subcubic Graphs

  • BOJAN MOHAR (a1)

Abstract

It is proved that the median eigenvalues of every connected bipartite graph G of maximum degree at most three belong to the interval [−1, 1] with a single exception of the Heawood graph, whose median eigenvalues are $\pm\sqrt{2}$ . Moreover, if G is not isomorphic to the Heawood graph, then a positive fraction of its median eigenvalues lie in the interval [−1, 1]. This surprising result has been motivated by the problem about HOMO-LUMO separation that arises in mathematical chemistry.

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[1] Gutman, I. and Polanski, O. E. (1986) Mathematical Concepts in Organic Chemistry, Springer.
[2] Fowler, P. W. and Pisanski, T. (2010) HOMO-LUMO maps for fullerenes. Acta Chim. Slov. 57 513517.
[3] Fowler, P. W. and Pisanski, T. (2010) HOMO-LUMO maps for chemical graphs. MATCH Commun. Math. Comput. Chem. 64 373390.
[4] Godsil, C. and Royle, G. (2001) Algebraic Graph Theory, Springer.
[5] Jaklič, G., Fowler, P. W. and Pisanski, T. (2012) HL-index of a graph. Ars Math. Contemp. 5 99105.
[6] Mohar, B. (2013) Median eigenvalues of bipartite planar graphs. MATCH Commun. Math. Comput. Chem. 70 7984.
[7] Mohar, B. (2015) Median eigenvalues and the HOMO-LUMO index of graphs. J. Combin. Theory Ser. B 112 7892.

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Median Eigenvalues of Bipartite Subcubic Graphs

  • BOJAN MOHAR (a1)

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