## Journal of Applied Mathematics

### The Distance Matrices of Some Graphs Related to Wheel Graphs

#### Abstract

Let $D$ denote the distance matrix of a connected graph $G$. The inertia of $D$ is the triple of integers (${n}_{+}\left(D\right), {n}_{0}\left(D\right), {n}_{-}\left(D\right)$), where ${n}_{+}\left(D\right)$, ${n}_{\mathrm{0}}\left(D\right)$, and ${n}_{-}\left(D\right)$ 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.

#### Article information

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

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394808026

Digital Object Identifier
doi:10.1155/2013/707954

Mathematical Reviews number (MathSciNet)
MR3067124

Zentralblatt MATH identifier
1271.05060

#### Citation

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. https://projecteuclid.org/euclid.jam/1394808026

#### References

• 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.