Abstract
The Wiener Index of a graph, known as the “sum of distances” of a connected graph, is the first topological index used in chemistry to sum the distances between all unordered pairs of vertices of a graph. In this paper, the lines of unit cells of the body-centered cubic grid are used. These graphs contain center points of the unit cells and other vertices, called border vertices. Closed formulae are obtained to calculate the sum of shortest distances between pairs of border vertices, between border vertices and centers and between pairs of centers. Based on these formulae, their sum, the Wiener Index of body-centered cubic grid with unit cells connected in a row graph is computed. Some relationships between formulae and integer sequences are also presented.
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References
Al-Kandari, A., Manuel, P., Rajasingh, I.: Wiener Index of Sodium Chloride and Benzenoid Structures. The Journal of Combinatorial Mathematics and Combinatorial Computing 79, 33–42 (2011)
Balaban, A.T., Mills, D., Ivanciuc, O., Basak, S.C.: Reverse Wiener indices. Croatica Chemica Acta 73, 923–941 (2000)
Bogdanov, B., Nikolić, S., Trinajstić, N.: On the three-dimensional Wiener number. Journal of Mathematical Chemistry 3, 299–309 (1989)
Bollobás, B.: Graph Theory: An Introductory Course. Springer, New York (1979)
Dobrynin, A.A., Gutman, I., Klavžar, S., Žigert, P.: Wiener index of hexagonal systems. Acta Appl. Math. 72, 247–294 (2002)
Kittel, C.: Introduction to Solid State Physics. Wiley, New York (2004)
Klavzar, S., Gutman, I.: Wiener number of vertex-weighted graphs and a chemical application. Discrete Applied Mathematics 80, 73–81 (1997)
Klette, R., Rosenfeld, A.: Digital geometry – geometric methods for digital picture analysis. Morgan Kaufmann (2004)
Knor, M., Skrekovski, R.: Wiener Index of generalized 4-stars and of their quadratic line graphs. Australasian Journal of Combinatorics 58, 119–126 (2014)
Manuel, P., Rajasingh, I., Arockiaraj, M.: Wiener and Szeged indices of Regular Tessellations. In: International Conference on Information and Network Technology (ICINT 2012). IPCSIT 37, pp. 210–214. IACSIT Press, Singapore (2012)
Mihalic, Z., Veljan, D., Amic, D., Nilkolic, S., Plavsic, D., Trianjstic, N.: The distance matrix in chemistry. J. Math. Chem. 11, 223–258 (1992)
Mohar, B., Pisanski, T.: How to compute the Wiener index of a graph. J. Math. Chem. 2, 267–277 (1988)
Nagy, B., Strand, R.: Non-Traditional Grids Embedded in Zn. International Journal of Shape Modeling 14, 209–228 (2008)
O’Keeffe, M.: Coordination sequences for lattices. Zeit. f. Krist. 210, 905–908 (1995)
Sloane, N.: On-Line Encyclopedia of Integer Sequences (OEIS), http://oeis.org/
Strand, R., Nagy, B.: Distances based on neighbourhood sequences in non-standard three-dimensional grids. Discrete Applied Mathematics 155, 548–557 (2007)
Strand, R., Nagy, B.: Path-based distance functions in n-dimensional generalizations of the face-and body-centered cubic grids. Discrete Applied Mathematics 157, 3386–3400 (2009)
Strand, R., Nagy, B., Borgefors, G.: Digital distance functions on three-dimensional grids. Theoretical Computer Science 412, 1350–1363 (2011)
Wiener, H.: Structural determination of paraffin boiling points. Journal of American Chemical Society 69, 17–20 (1947)
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Mujahed, H., Nagy, B. (2015). Wiener Index on Lines of Unit Cells of the Body-Centered Cubic Grid. In: Benediktsson, J., Chanussot, J., Najman, L., Talbot, H. (eds) Mathematical Morphology and Its Applications to Signal and Image Processing. ISMM 2015. Lecture Notes in Computer Science(), vol 9082. Springer, Cham. https://doi.org/10.1007/978-3-319-18720-4_50
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DOI: https://doi.org/10.1007/978-3-319-18720-4_50
Publisher Name: Springer, Cham
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