| Abstrakt: |
AbstractThe Wiener index of a connected graph Gis W(G)=∑{u,v}⊆V(G)dG(u,v). In this paper, we obtain the Wiener index of H-generalized join of graphs G1,G2,…,Gk. As a consequence, we obtain some earlier known results in [Alaeiyan et al. in Aust. J. Basic Appl. Sci. (2011) 5(12): 145–152; Yeh et al. in Discrete Math. (1994) 135: 359–365] and we also obtain the Wiener index of the generalized corona product of graphs. We further show that the ideal-based zero-divisor graph ΓI(R)is a H-generalized join of complete graphs and totally disconnected graphs. As a result, we find the Wiener index of the ideal-based zero-divisor graph ΓI(R)and we deduce some of the main results in [Selvakumar et al. in Discrete Appl. Math. (2022) 311: 72–84]. Moreover, we show that W(ΓI(Zn))is a quadratic polynomial in n, where Znis the ring of integers modulo nand we calculate the exact value of the Wiener index of ΓNil(R)(R), where Nil(R) is nilradical of R. Furthermore, we give a Python program for computing the Wiener index of ΓI(Zn)if Iis an ideal of Zngenerated by pr, where pris a proper divisor of n, pis a prime number and ris a positive integer with r≥2. |
