Rocky Mountain Journal of Mathematics

The minimum matching energy of bicyclic graphs with given girth

Hong-Hai Li and Li Zou

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The matching energy of a graph was introduced by Gutman and Wagner in 2012 and defined as the sum of the absolute values of zeros of its matching polynomial. Let $\theta (r,s,t)$ be the graph obtained by fusing two triples of pendant vertices of three paths $P_{r+2}$, $P_{s+2}$ and $P_{t+2}$ to two vertices. The graph obtained by identifying the center of the star $S_{n-g}$ with the degree~3 vertex $u$ of $\theta (1,g-3,1)$ is denoted by $S_{n-g}(u)\theta (1,g-3,1)$. In this paper, we show that, $S_{n-g}(u)\theta (1,g-3,1)$ has minimum matching energy among all bicyclic graphs with order $n$ and girth $g$.

Article information

Rocky Mountain J. Math., Volume 46, Number 4 (2016), 1275-1291.

First available in Project Euclid: 19 October 2016

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Zentralblatt MATH identifier

Primary: 05C35: Extremal problems [See also 90C35] 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.)

Bicyclic graph matching energy girth


Li, Hong-Hai; Zou, Li. The minimum matching energy of bicyclic graphs with given girth. Rocky Mountain J. Math. 46 (2016), no. 4, 1275--1291. doi:10.1216/RMJ-2016-46-4-1275.

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  • L. Chen, J. Liu and Y. Shi, Matching energy of unicyclic and bicyclic graphs with a given diameter, Complexity 21 (2015), 224–238.
  • L. Chen and Y. Shi, The maximal matching energy of tricyclic graphs, MATCH Comm. Math. Comp. Chem. 73 (2015), 105–120.
  • D.M. Cvetković, M. Doob, I. Gutman and A. Torgašev, Recent results in the theory of graph spectra, North-Holland, Amsterdam, 1988.
  • H. Deng, The smallest Hosoya index in $(n, n+ 1)$-graphs, J. Math. Chem. 43 (2008), 119–133.
  • I. Gutman, The matching polynomial, MATCH Comm. Math. Comp. Chem. 6 (1979), 75–91.
  • I. Gutman and D.M. Cvetković, Finding tricyclic graphs with a maximal number of matchings–Another example of computer aided research in graph theory, Publ. Inst. Math. Nouv. 35 (1984), 33–40.
  • I. Gutman and S. Wagner, The matching energy of a graph, Discr. Appl. Math. 160 (2012), 2177–2187.
  • I. Gutman and F. Zhang, On the ordering of graphs with respect to their matching numbers, Discr. Appl. Math. 15 (1986), 25–33.
  • G. Huang, M. Kuang and H. Deng, Extremal graph with respect to matching energy for a random polyphenyl chain, MATCH Comm. Math. Comp. Chem. 73 (2015), 121–131.
  • S. Ji, X. Li and Y. Shi, Extremal matching energy of bicyclic graphs, MATCH Comm. Math. Comp. Chem. 70 (2013), 697–706.
  • S. Ji and H. Ma, The extremal matching energy of graphs, Ars Combin. 115 (2014), 343–355.
  • S. Ji, H. Ma and G. Ma, The matching energy of graphs with given edge connectivity, J. Inequal. Appl. 1 (2015), 1–9.
  • X. Li, Y. Shi and I. Gutman, Graph energy, Springer, New York, 2012.
  • X. Li, Y. Shi, M. Wei and J. Li, On a conjecture about tricyclic graphs with maximal energy, MATCH Comm. Math. Comp. Chem. 72 (2014), 183–214.
  • H. Li, B. Tan and L. Su, On the signless Laplacian coefficients of unicyclic graphs, Linear Alg. Appl. 439 (2013), 2008–2009.
  • S. Li and W. Yan, The matching energy of graphs with given parameters, Discr. Appl. Math. 162 (2014), 415–420.
  • H. Li, Y. Zhou and L. Su, Graphs with extremal matching energies and prescribed parameters, MATCH Comm. Math. Comp. Chem. 72 (2014), 239–248.
  • R. Sun, Z. Zhu and L. Tan, On the Merrifield-Simmons index and Hosoya index of bicyclic graphs with a given girth, Ars Combin. 103 (2012), 465–478.
  • K. Xu, K.C. Das and Z. Zheng, The minimal matching energy of $(n,m)$-graphs with a given matching number, MATCH Comm. Math. Comp. Chem. 73 (2015), 93–104.