Extremal Matching Energy of Complements of Trees

Tingzeng Wu; Weigen Yan; Heping Zhang

Discussiones Mathematicae Graph Theory (2016)

  • Volume: 36, Issue: 3, page 505-521
  • ISSN: 2083-5892

Abstract

top
Gutman and Wagner proposed the concept of the matching energy which is defined as the sum of the absolute values of the zeros of the matching polynomial of a graph. And they pointed out that the chemical applications of matching energy go back to the 1970s. Let T be a tree with n vertices. In this paper, we characterize the trees whose complements have the maximal, second-maximal and minimal matching energy. Furthermore, we determine the trees with edge-independence number p whose complements have the minimum matching energy for p = 1, 2, . . . , [n/2]. When we restrict our consideration to all trees with a perfect matching, we determine the trees whose complements have the second-maximal matching energy.

How to cite

top

Tingzeng Wu, Weigen Yan, and Heping Zhang. "Extremal Matching Energy of Complements of Trees." Discussiones Mathematicae Graph Theory 36.3 (2016): 505-521. <http://eudml.org/doc/285395>.

@article{TingzengWu2016,
abstract = {Gutman and Wagner proposed the concept of the matching energy which is defined as the sum of the absolute values of the zeros of the matching polynomial of a graph. And they pointed out that the chemical applications of matching energy go back to the 1970s. Let T be a tree with n vertices. In this paper, we characterize the trees whose complements have the maximal, second-maximal and minimal matching energy. Furthermore, we determine the trees with edge-independence number p whose complements have the minimum matching energy for p = 1, 2, . . . , [n/2]. When we restrict our consideration to all trees with a perfect matching, we determine the trees whose complements have the second-maximal matching energy.},
author = {Tingzeng Wu, Weigen Yan, Heping Zhang},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {matching polynomial; matching energy; Hosoya index; energy},
language = {eng},
number = {3},
pages = {505-521},
title = {Extremal Matching Energy of Complements of Trees},
url = {http://eudml.org/doc/285395},
volume = {36},
year = {2016},
}

TY - JOUR
AU - Tingzeng Wu
AU - Weigen Yan
AU - Heping Zhang
TI - Extremal Matching Energy of Complements of Trees
JO - Discussiones Mathematicae Graph Theory
PY - 2016
VL - 36
IS - 3
SP - 505
EP - 521
AB - Gutman and Wagner proposed the concept of the matching energy which is defined as the sum of the absolute values of the zeros of the matching polynomial of a graph. And they pointed out that the chemical applications of matching energy go back to the 1970s. Let T be a tree with n vertices. In this paper, we characterize the trees whose complements have the maximal, second-maximal and minimal matching energy. Furthermore, we determine the trees with edge-independence number p whose complements have the minimum matching energy for p = 1, 2, . . . , [n/2]. When we restrict our consideration to all trees with a perfect matching, we determine the trees whose complements have the second-maximal matching energy.
LA - eng
KW - matching polynomial; matching energy; Hosoya index; energy
UR - http://eudml.org/doc/285395
ER -

References

top
  1. [1] J. Aihara, A new definition of Dewar-type resonance energies, J. Amer. Chem. Soc. 98 (1976) 2750-2758. doi:10.1021/ja00426a013[Crossref] 
  2. [2] L. Chen and Y. Shi, The maximal matching energy of tricyclic graphs, MATCH Commun. Math. Comput. Chem. 73 (2015) 105-119. 
  3. [3] L. Chen, J. Liu and Y. Shi, Matching energy of unicyclic and bicyclic graphs with a given diameter , Complexity 21 (2015) 224-238. doi:10.1002/cplx.21599[Crossref][WoS] 
  4. [4] D. Cvetković, M. Doob, I. Gutman and A. Torgašev, Recent Results in the Theory of Graph Spectra (North-Holland, Amsterdam, 1988). Zbl0634.05054
  5. [5] M.J.S. Dewar, The Molecular Orbital Theory of Organic Chemistry (McGraw-Hill, New York, 1969). 
  6. [6] E.J. Farrell, An introduction to matching polynomials, J. Combin. Theory Ser. B 27 (1979) 75-86. doi:10.1016/0095-8956(79)90070-4[Crossref] 
  7. [7] C.D. Godsil, Algebraic Combinatorics (Chapman and Hall, New York, 1993). 
  8. [8] C.D. Godsil and I. Gutman, On the theory of the matching polynomial , J. Graph Theory 5 (1981) 137-144. doi:10.1002/jgt.3190050203[Crossref] 
  9. [9] I. Gutman, The matching polynomial , MATCH Commun. Math. Comput. Chem. 6 (1979) 75-91. Zbl0436.05053
  10. [10] I. Gutman and S. Wagner, The matching energy of a graph, Discrete Appl. Math. 160 (2012) 2177-2187. doi:10.1016/j.dam.2012.06.001[WoS][Crossref] Zbl1252.05120
  11. [11] I. Gutman, The energy of a graph: old and new results, in: Algebraic Combina- torics and Applications, A. Betten, A. Kohnert, R. Laue, A. Wassermann (Ed(s)), (Springer-Verlag, Berlin, 2001) 196-211. doi:10.1007/978-3-642-59448-9 13[Crossref] Zbl0974.05054
  12. [12] I. Gutman, X. Li and J. Zhang, Graph energy, in: Analysis of Complex Networks From Biology to Linguistics, M. Dehmer, F. Emmert-Streib (Ed(s)), (Wiley-VCH, Weinheim, 2009) 145-174. doi:10.1002/9783527627981.ch7[Crossref] 
  13. [13] S. Ji, X. Li and Y. Shi, Extremal matching energy of bicyclic graphs, MATCH Commun. Math. Comput. Chem. 70 (2013) 697-706. Zbl1299.05220
  14. [14] H. Li, Y. Zhou and L. Su, Graphs with extremal matching energies and prescribed parameters, MATCH Commun. Math. Comput. Chem. 72 (2014) 239-248. 
  15. [15] S. Li and W. Yan, The matching energy of graphs with given parameters, Discrete Appl. Math. 162 (2014) 415-420. doi:10.1016/j.dam.2013.09.014[WoS][Crossref] Zbl1300.05162
  16. [16] X. Li, Y. Shi and I. Gutman, Graph Energy (Springer, New York, 2012). doi:10.1007/978-1-4614-4220-2[Crossref] 
  17. [17] L. Lovász, Combinatorial Problems and Exercises, Second Edition (Budapest, Akad´emiai Kiad´o, 1993). Zbl0785.05001
  18. [18] D.B. West, Introduction to Graph Theory, Second Edition (Pearson Education, Singapore, 2001). 
  19. [19] T. Wu, On the maximal matching energy of graphs, J. East China Norm. Univ. 1 (2015) 136-141.  
  20. [20] K. Xu, Z. Zheng and K.C. Das, Extremal t-apex trees with respect to matching energy, Complexity (2015), in press. doi:10.1002/cplx.21651[Crossref] 
  21. [21] K. Xu, K.C. Das and Z. Zheng, The minimum matching energy of (n,m)-graphs with a given matching number , MATCH Commun. Math. Comput. Chem. 73 (2015) 93-104. 
  22. [22] W. Yan, Y. Yeh and F. Zhang, Ordering the complements of trees by the number of maximum matchings, J. Quan. Chem. 1055 (2005) 131-141. doi:10.1002/qua.20688[Crossref] 

NotesEmbed ?

top

You must be logged in to post comments.