On ABS index of unicyclic graphs with fixed diameter

Given a simple connected graph G = (V(G), E(G)), the atom-bond sum-connectivity (ABS) index is defined as ABS(G)=?uv?E(G) ?d(u)+d (v)- 2/d(u)+d(v)=?uv?E(G)?1-2/d(u)+ d(v)', where d(u) and d(v) are the degrees of u, v ? V(G), respectively. In this paper, let Un,? be a set of all unicyclic graphs of order n with diameter ?. Firstly, we present the minimum ABS index of G ? Un,? with ? ? 2. We also determine the maximum ABS index of G ? Un,? with ? ? 4. Finally, the corresponding extremal graphs with the sharp upper and lower bounds have been characterized, respectively.