Sharp Bounds on Merrifield-Simmons Index of the Generalized \( \theta \)-Graph

For a graph \( G \), the Merrifield-Simmons index \( i(G) \) is defined as the total number of its independent sets. In this paper, we determine sharp upper and lower bounds on Merrifield-Simmons index of generalized \( \theta \)-graph, which is obtained by subdividing the edges of the multigraph consisting of \( k \) parallel edges, denoted by \( \theta(r_1, r_2, \ldots, r_k) \). The corresponding extremal graphs are also characterized.