Volume 11 , Issue 1 , PP: 25-34, 2024 | Cite this article as | XML | Html | PDF | Full Length Article
Lee Xu, Taher 1 * , Ahmed Jubbori 2
Doi: https://doi.org/10.54216/GJMSA.0110103
Let u and v be any two distinct vertices in a connected graph G. A container C(u,v) is a set of internally disjoint u - v paths. The width of C(u,v) is denoted by w or w(C(u,v)), it is equal to , and the length of is the length of the longest u – v path in C(u,v). Then, for a given positive integer w, the width distance between any two distinct vertices u and v in a connected graph G is define by:dw (u,v)=min/(C(u,v)) l(C(u,v)) , where the minimum is taken over all containers C(u, v) of width w.In this paper, we find the Hosoya polynomials and Wiener indices of the join of two special graphs such as bipartite complete graphs, paths, cycles, star graphs and wheel graphs with respect to the width distance.
Connected graph , Path, Width , Hosoya polynomial
[1] A. Ali and A. S. Aziz; (2007), "w- Wiener Polynomials of the width Distance of some special graphs", Al-Rafiden J. vol. 4, No. 2, pp. 103-124.
[2] E. Sagan, Y-N. Yeh, and P. Zhang; (1996), "The Wiener Polynomial of a Graph", Intern. J. of Quantum Chemistry, Vol. 60, pp. 959-969.
[3] D.F. Hsu; (1994), "On Container Width and Length in Graphs, Groups, and Networks", IEICE Transactions on Fundamentals of Electronics communications and computer sciences", Vol. E77-A, No.4, pp. 668-680.
[4] R. Diestel, (2005). "Graph Theory", Springer–Verlag Heidelberg, New York.
[5] Gagnon, A. Hassler, J. Huang, A. Krim-Yee, F.M. Inerney, A.M. Zacarias, B. Seamone, V. Virgile, A method for eternally dominating strong grids, Discrete Math Theor Comput Sci., 22 (2020), 1j+. https://doi.org/10.23638/DMTCS-22-1-8
[6] Pradhan, S. Banerjee, L. Jia-Bao. Perfect Italian domination in graphs: Complexity and algorithms, Discrete. Appl. Math., 319 (2022), 271-295. https://doi.org/10.1016/j.dam.2021.08.020
[7] J. Varghese, A. Lakshmanan, Perfect Italian Domination Number of Graphs, Palest. J. Math., 12 (2023), 158-168. Available from: https://pjm.ppu.edu/paper/1259-perfect-italian-domination-number-graphs
[8] F. Romeo, Chordal circulant graphs and induced matching number. Discrete. Math., 343 (2020), 111947. https://doi.org/10.1016/j.disc.2020.111947