aspg-33-3089-1792223347 20261017074907 American Scientific Publishing Group admin@americaspg.com American Scientific Publishing Group Galoitica: Journal of Mathematical Structures and Applications GJMSA 2834-5568 2024 11 1 Lee Xu, Taher University of Chinese Academy of Sciences, CAS, Mathematics Department, Beijing, China Ahmed Jubbori Computer Techniques Engineering Department, Al-Mustaqbal University, Babil, Iraq 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 m:val="|"/> 10.0pt;mso-bidi-font-size:10.0pt;font-family:"Cambria Math",serif; mso-ascii-font-family:"Cambria Math";mso-fareast-font-family:Calibri; mso-hansi-font-family:"Cambria Math";mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:major-bidi;font-style:italic;mso-bidi-font-style:normal'> style='mso-bidi-font-style:normal'> "Cambria Math",serif;mso-fareast-font-family:Calibri;mso-bidi-font-family: "Times New Roman";mso-bidi-theme-font:major-bidi'>C(u,v) , 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. 2024 25 34 3089 https://creativecommons.org/licenses/by/4.0/ 10.54216/GJMSA.0110103 https://americaspg.com/journal/33/article/3089 https://americaspg.com/storage/91722627928.pdf