The extremal θ-graphs with respect to Hosoya index and Merrifield-Simmons index

For a graph G, the Hosoya index and the Merrifield-Simmons index are defined as the total number of its matchings and the total number of its independent sets, respectively. In this paper, we obtain the smallest and the largest Hosoya index and Merrifield-Simmons index of theta-graphs, which are obtained by subdividing the edges of the multigraph consisting of 3 parallel edges, denoted by theta(r, s, t). At the same time, we characterize the corresponding extremal graphs. Our results show that the graphs of the smallest Merrifield-Simmons index do not coincide with those of the largest Hosoya index among theta-graphs. Surprisedly, they attain their lower bound for at the same graph theta(0, 1, n - 3).