The Hosoya index and the Merrifield–Simmons index

In this article, we give sharp bounds on the Hosoya index and the Merrifield–Simmons index for connected graphs of fixed size. As a consequence, we determine all connected graphs of any fixed order and size which maximize the Merrifield–Simmons index. Sharp lower bounds on the Hosoya index are known for graphs of order n and size m∈[n-1,2n-3]∪n-12,n2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\in [n-1,2n-3]\cup \left( {n-1\atopwithdelims ()2},{n\atopwithdelims ()2}\right] $$\end{document}; while sharp upper bounds were only known for graphs of order n and size m≤n+2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\le n+2$$\end{document}. We give sharp upper bounds on the Hosoya index for dense graphs with m≥n2-2n/3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\ge {n\atopwithdelims ()2}-2n/3$$\end{document}. Moreover, all extreme graphs are also determined.