Abstract
Let \(\fancyscript{T}_{n,\gamma }\) be the collection of all \(n\)-vertex trees with domination number \(\gamma \). In this paper, the first-, second- and third-smallest Laplacian permanents of trees in \(\fancyscript{T}_{n,\gamma }\) are determined, respectively. Moreover, the corresponding extremal graphs are characterized.
Similar content being viewed by others
References
Bapat, R.B.: A bound for the permanent of the Laplacian matrix. Linear Algebr. Appl. 74, 219–223 (1986)
Bregman, L.M.: Some properties of nonnegative matrices and their permanents. Sov. Math. Dokl. 14(4), 945–949 (1973)
Brualdi, R.A., Goldwasser, J.L.: Permanent of the Laplacian matrix of tree and bipartite graphs. Discret. Math. 48, 1–12 (1984)
Brualdi, R.A., Goldwasser, J.L., Michael, T.S.: Maximum permanents of matrices of zeros and ones. J. Combin. Theory Ser. A 47, 207–245 (1988)
Cvetković, D., Simić, S.: Towards a spectral theory of graphs based on the signless Laplacian. I. Publ. Inst. Math. (Beograd) 85(99), 19–33 (2009)
Cvetković, D., Simić, S.: Towards a spectral theory of graphs based on the signless Laplacian. II. Linear Algebr. Appl. 432, 2257–2272 (2010)
Cvetković, D., Simić, S.: Towards a spectral theory of graphs based on the signless Laplacian. III. Appl. Anal. Discrete Math. 4, 156–166 (2010)
Codenotti, B., Crepsi, V., Resta, G.: On the permanent of certain \((0, 1)\) Toeplitz matrices. Linear Algebr. Appl. 267, 65–100 (1997)
Farrell, E.J., Kennedy, J.W., Quintas, L.V.: Permanents and determinants of graphs: a cycle polynomial approach. J. Combin. Math. Combin. Comput. 32, 129–137 (2000)
Minc, H.: Bounds for permanents of nonnegative matrices. Proc. Edinb. Math. Soc. 16, 233–237 (1968)
Minc, H.: Upper bounds for permanents of \((0, 1)\)-matrices. Bull. Am. Math. Soc. 69, 789–791 (1963)
Fink, J.F., Jocobson, M.S., Kinch, L.K., Roberts, J.: On graphs having domination number half their order. Period. Math. Hungar. 16, 287–293 (1985)
Frucht, R., Harary, F.: On the corona of two graphs. Aequ. Math. 4, 322–324 (1970)
Geng, X.Y., Hu, X., Li, S.C.: Further results on permanental bounds for the Laplacian matrix of trees. Linear Multilinear Algebr. 58, 571–587 (2010)
Li, S.C., Zhang, L.: Permanental bounds for the signless Laplacian matrix of a unicyclic graph with diameter \(d\). Graphs Combin. 28, 531–546 (2012)
Li, S.C., Wang, S.J.: Further analysis on the total number of subtrees of trees. Electron. J. Combin. 19(4), P48 (2012)
Li, S.C., Zhang, L.: Permanental bounds for the signless Laplacian matrix of bipartite graphs and unicyclic graphs. Linear Multilinear Algebr. 59, 145–158 (2011)
Li, S.C., Li, Y., Zhang, X.X.: Edge-grafting theorems on permanents of the laplacian matrices of graphs and their applications. Electron. J. Linear Algebr. 26, 28–48 (2013)
Merris, R.: The Laplacian permanental polynomial for trees. Czechoslov. Math. J. 32(107), 397–403 (1982)
Merris, R., Rebman, K.R., Watkins, W.: Permanental polynomials of graph. Linear Algebr. Appl. 38, 273–288 (1981)
Minc, H.: Permanents. Addison-Wesley, Reading, MA (1978)
Ore, O.: Theory of graphs. In: Amer. Math. Soc. Colloq. Publ., 38 (1962)
Samorodnitsky, A.: An upper bound for permanents of nonnegative matrices. J. Combin. Theory Ser. A 115, 279–292 (2008)
Soules, G.W.: Extending the Minc-Bregman upper bound for the permanent. Linear Multilinear Algebr. 47, 77–91 (2000)
Soules, G.W.: New permanental upper bounds for nonnegative matrices. Linear Multilinear Algebr. 51, 319–337 (2003)
Soules, G.W.: Permanental bounds for nonnegative matrices via decomposition. Linear Algebr. Appl. 394, 73–89 (2005)
Wanless, I.M.: Maximizing the permanent at the complementary permanent of \((0, 1)\)-matrices with constant line sum. Discret. Math. 205, 191–205 (1999)
Xu, B., Cockayne, E.J., Haynes, T.W., Hedetniemi, S.T., Zhou, S.C.: Extremal graphs for inequalities involving domination parameters. Discret. Math. 216, 1–10 (2000)
Zhang, F.Z.: An analytic approach to a permanent conjecture. Linear Algebr. Appl. 438, 1570–1579 (2013)
Acknowledgments
Financially supported by the National Natural Science Foundation of China (Grant Nos. 11271149, 11371062), the Program for New Century Excellent Talents in University (Grant No. NCET-13-0817) and the Special Fund for Basic Scientific Research of Central Colleges (Grant No. CCNU13F020).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Geng, X., Hu, S. & Li, S. Permanental Bounds of the Laplacian Matrix of Trees with Given Domination Number. Graphs and Combinatorics 31, 1423–1436 (2015). https://doi.org/10.1007/s00373-014-1451-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-014-1451-z