ON THE TOPOLOGICAL INDICES OF WEAKLY ZERO-DIVISOR GRAPH

The Wiener index of a connected graph $G$ is defined as $W(G)=\linebreak \sum_{u, v} d_{G}(u, v)$, where $d_{G}(u, v)$ denotes the distance between the vertices $u$ and $v$ and the sum runs over all unordered pairs of vertices. The Harary index of a connected graph $G$ is the sum of reciprocal of distances between all pairs of vertices i.e. $H(G)= \sum \frac{1}{d_G(u,v) }$, where the summation runs over all unordered pairs u and v of vertices of $G$. The weakly zero-divisor graph of a commutative ring $R$ ( $\Gamma' (R)$) is a simple undirected graph with vertex set as the set of all non zero zero-divisors of $R$ and the distinct vertices $x$ and $y$ are joined by an edge if and only if there exist $r \in ann(x)$ and $s \in ann(y)$ such that $rs = 0$. In this paper, we find the Wiener index and Harary index of $\Gamma'(R)$ where $R$ is the ring of integers modulo $n$.