The Expected Values of Merrifield-Simmons Index in Random Phenylene Chains

Document Type : Research Paper


1 Xinjiang Normal University

2 Department of Mathematics, Xinjiang Normal University, Urumqi, Xinjiang 830054, P.R.China

3 Xinjiang University

4 Xinjiang normal University


The Merrifield-Simmons index of a graph G is the number of independent sets in G. In this paper, we give exact formulae for the expected value of the Merrifield-Simmons index of random phenylene chains by means of auxiliary graphs.



    1. H. Abdo, R‎. ‎E‎. ‎Merrifield‎ and ‎H‎. ‎E‎. ‎Simmons‎, ‎The structure of molecular topological spaces, Theor. Chim. Acta‎. ‎55 (1980) 55-57.‎ ‎
    2.  R‎. ‎E‎. ‎Merrifield‎ and ‎H‎. ‎E‎. ‎Simmons‎, ‎Enumeration of strucure-sensitive graphical subset‎: ‎theory‎, ‎Proc. Natl. Acad. Sci. USA‎ 78 (1981) 692-695‎. ‎‎ ‎
    3.  R‎. ‎E‎. ‎Merrifield and ‎H‎. ‎E‎. ‎Simmons‎, ‎Enumeration of strucure-sensitive graphical subset‎: ‎calculation. Proc. Natl. Acad. Sci. USA 78 (1981) 1329-1332‎. ‎‎ ‎
    4. R‎. ‎E‎. ‎Merrifield‎ and ‎H‎. ‎E‎. ‎Simmons‎, ‎Topology of bonding in  systems‎, Proc. Natl. Acad. Sci. USA‎  82 (1985) 1-3.‎ ‎
    5. R‎. ‎E‎. ‎Merrifield and ‎H‎. ‎E‎. ‎Simmons‎, ‎Mathematical description of molecular structure; molecular topolygoy, ‎Proc. Natl. Acad. Sci. USA‎ 74 (1977) 2616-2619.‎ ‎
    6. S‎. ‎Wagner and ‎I‎. ‎Gutman‎, ‎Maxima and minima of the Hosoya index and Merrifield-Simmons index‎, ‎Acta. Appl. Math‎.112 (2010) 323-346.‎ ‎
    7. E‎. ‎O‎. ‎D. Andriatiana‎, ‎Energy‎, ‎Hosoya index and Merrifield-Simmons index of trees with prescribed degree sequence‎, ‎Discret. Appl. Math‎.161 (2013) 724-741.‎ ‎
    8. Y‎. ‎Liu‎, ‎W‎. ‎Zhuang‎ and ‎Z‎. ‎Liang, Largest Hosoya index and smallest Merrifield-Simmons index in tricyclic graph‎, ‎MATCH Commun. Math. Comput. Chem‎.73 (2015) 195-224.‎ ‎
    9. Z‎. ‎Zhu‎, ‎C‎. ‎Yuan‎, ‎E‎. ‎O‎. ‎D. Andriantiana‎ and ‎S‎. ‎Wagner‎, ‎Graphs with maximal Hosoya index and minimal Merrifield-Simmons index, ‎Discret Math‎.329 (2014) 77-87.‎ ‎
    10. X‎. ‎Chen‎, ‎B‎. ‎Zhao‎ and ‎P. Zhao‎, ‎Six-member ring spiro chains with extremal Merrifield-Simmons index and Hosoya index‎, ‎MATCH Common. Math. Comput. Chem‎.62 (2009) 657-665.‎ ‎
    11. X‎. ‎Li‎, ‎X‎. ‎Yang‎, ‎G‎. ‎Wang‎ and ‎R‎. ‎Hu‎, ‎Hosoya Polynomials of General Spiro Hexagonal Chains‎, ‎Faculty of Science and Mathematics‎, ‎University of Niš, ‎Seria‎. ‎28 (1) (2014) 221-215.‎ ‎
    12. H‎. ‎Deng‎, ‎Wiener indices of spiro and polyphenyl hexagonal chains‎, ‎Math. Comput. Model‎.55 (2012) 634-644.‎ ‎
    13. H‎. ‎Deng‎ and ‎Z‎. ‎Tang‎, ‎Kirchhoff indices of spiro and polyphenyl hexagonal chains‎, ‎Util. Math‎. ‎95 (2014) 113-128.‎ ‎
    14. Y‎. ‎Bai‎, ‎B‎. ‎Zhao‎ and ‎P‎. ‎Zhao‎, ‎Extermal Merrifield-Simmons index and Hosoya index of polyphenyl chains‎, ‎MATCH Commum Math Comput Chem‎. ‎62 (2019) 649-656.‎ ‎
    15. T‎. ‎Došlić‎ and ‎M‎. ‎S‎. ‎Litz, Matchings and independent sets in polyphenyl chains, ‎MATCH Common Math Comput Chem‎. ‎67 (2012) 313-330.‎ ‎
    16. B‎. ‎Liu‎, ‎H‎. ‎Bian‎ and ‎H‎. ‎Yu, The expected values of Hosoya index and Merrifield-Simmons index in  random spiro chains, ‎Ars Combinatoria‎ ‎147 (2019) 323-336.‎ ‎