Journal of Applied Mathematics

The Distance Matrices of Some Graphs Related to Wheel Graphs

Xiaoling Zhang and Chengyuan Song

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Let D denote the distance matrix of a connected graph G. The inertia of D is the triple of integers (n+(D), n0(D), n-(D)), where n+(D), n0(D), and n-(D) denote the number of positive, 0, and negative eigenvalues of D, respectively. In this paper, we mainly study the inertia of distance matrices of some graphs related to wheel graphs and give a construction for graphs whose distance matrices have exactly one positive eigenvalue.

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J. Appl. Math., Volume 2013 (2013), Article ID 707954, 5 pages.

First available in Project Euclid: 14 March 2014

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Zhang, Xiaoling; Song, Chengyuan. The Distance Matrices of Some Graphs Related to Wheel Graphs. J. Appl. Math. 2013 (2013), Article ID 707954, 5 pages. doi:10.1155/2013/707954.

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  • D. H. Rouvray, “The role of the topological distance matrix in chemistry,” in Mathematical and Computational Concepts in Chemistry, N. Trinajstid, Ed., pp. 295–306, Ellis Harwood, Chichester, UK, 1986.
  • W. Brostow, D. M. McEachern, and S. Petrez- Guiterrez, “Pressure second virial coefficients of hydrocarbons, fluorocarbons, and their mixtures: Interactions of walks,” Journal of Chemical Physics, vol. 71, pp. 2716–2722, 1979.
  • B. D. McKay, “On the spectral characterisation of trees,” Ars Combinatoria, vol. 3, pp. 219–232, 1977.
  • D. W. Bradley and R. A. Bradley, “String edits and macromolecules,” in Time Wraps, D. Sankoff and J. B. Kruskal, Eds., Chapter 6, Addison-Wesley, Reading, Mass, USA, 1983.
  • J. P. Boyd and K. N. Wexler, “Trees with structure,” Journal of Mathematical Psychology, vol. 10, pp. 115–147, 1973.
  • R. L. Graham and L. Lovász, “Distance matrix polynomials of trees,” in Theory and Applications of Graphs, vol. 642 of Lecture Notes in Mathematics, pp. 186–190, 1978.
  • M. S. Waterman, T. F. Smith, and H. I. Katcher, “Algorithms for restriction map comparisons,” Nucleic Acids Research, vol. 12, pp. 237–242, 1984.
  • R. L. Graham and H. O. Pollak, “On the addressing problem for loop switching,” The Bell System Technical Journal, vol. 50, pp. 2495–2519, 1971.
  • R. Bapat, S. J. Kirkland, and M. Neumann, “On distance matrices and Laplacians,” Linear Algebra and Its Applications, vol. 401, pp. 193–209, 2005.
  • K. Balasubramanian, “Computer generation of distance polynomials of graphs,” Journal of Computational Chemistry, vol. 11, no. 7, pp. 829–836, 1990.
  • D. M. Cvetković, M. Doob, I. Gutman, and A. Torgašev, Recent Results in the Theory of Graph Spectra, vol. 36, North-Holland Publishing, Amsterdam, The Netherlands, 1988.
  • H. S. Ramane, D. S. Revankar, I. Gutman, and H. B. Walikar, “Distance spectra and distance energies of iterated line graphs of regular graphs,” Institut Mathématique, vol. 85, pp. 39–46, 2009.
  • X. Zhang and C. Godsil, “The inertia of distance matrices of some graphs,” Discrete Mathematics, vol. 313, no. 16, pp. 1655–1664, 2013.
  • D. M. Cvetković, M. Doob, and H. Sachs, Spectra of Graphs, vol. 87, Academic Press, New York, NY, USA, 1980, Theory and application.
  • R. L. Graham, A. J. Hoffman, and H. Hosoya, “On the distance matrix of a directed graph,” Journal of Graph Theory, vol. 1, no. 1, pp. 85–88, 1977.