The Wiener Index and Related Indices of the Zero Divisor Graphs of Certain Classes of Completely Primary Finite Rings

Abstract

Studies on zero divisor graphs of completely primary nite rings R have been extensively done from Galois rings to other rings whose sets of zero divisors Z(R) coincides with the Jacobson radical J (R). The studies have focused on graph geometric properties such as the girth, clique number, chromatic number and diameter among others. Some ndings are also evident on matrices of zero divisor graphs on certain classes of rings. The classes of completely primary nite rings considered in the various studies are square radical zero, cube radical zero and power four radical zero. In this paper we have advanced the study on zero divisor graph Γ(R) of completely primary nite rings by investigating the Wiener Index and its invariants such as the Average disorder number and Distance index. Further, we analyse the binding number and some bounds on the Zagreb indices of the rings satisfying the conditions (Z(R))3 = (0) and (Z(R))2 ≠ (0), (Z(R))4 = (0) and (Z(R))3 ≠ (0).