It is well-known that any graph has all real eigenvalues and a graph is bipartite if and only if its spectrum is symmetric with respect to the origin. We are interested in finding whether the permanental roots of a bipartite graph G have symmetric property as the spectrum of G. In this note, we show that the permanental roots of bipartite graphs are symmetric with respect to the real and imaginary axes. Furthermore, we prove that any graph has no negative real permanental root, and any graph containing at least one edge has complex permanental roots.
@article{bwmeta1.element.doi-10_7151_dmgt_1704,
author = {Heping Zhang and Shunyi Liu and Wei Li},
title = {A Note on the Permanental Roots of Bipartite Graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {34},
year = {2014},
pages = {49-56},
zbl = {1292.05142},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1704}
}
Heping Zhang; Shunyi Liu; Wei Li. A Note on the Permanental Roots of Bipartite Graphs. Discussiones Mathematicae Graph Theory, Tome 34 (2014) pp. 49-56. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1704/
[1] B. Anderson, J. Jackson and M. Sitharam, Descartes’ rule of signs revisited, Amer. Math. Monthly 105 (1998) 447-451. doi:10.2307/3109807[Crossref] | Zbl 0913.12001
[2] F. Belardo, V.D. Filippis and S.K. Simić, Computing the permanental polynomial of matrix from a combinatorial viewpoint , MATCH Commun. Math. Comput. Chem. 66 (2011) 381-396. | Zbl 1289.05268
[3] G.D. Birkhoff and D.C. Lewis, Chromatic polynomials, Trans. Amer. Math. Soc. 60 (1946) 355-451. doi:10.2307/1990348[Crossref]
[4] M. Borowiecki, On spectrum and per-spectrum of graphs, Publ. Inst. Math. (Beograd) 38 (1985) 31-33. | Zbl 0581.05036
[5] M. Borowiecki and T. Jóźwiak, A note on characteristic and permanental polynomials of multigraphs, in: Graph Theory, Borowiecki, Kennedy and Syslo (Ed(s)), (Berlin, Springer-Verlag, 1983) 75-78. | Zbl 0525.05046
[6] M. Borowiecki and T. Jóźwiak, Computing the permanental polynomial of a multigraph, Discuss. Math. V (1982) 9-16. | Zbl 0514.05044
[7] G.G. Cash, The permanental polynomial , J. Chem. Inf. Comput. Sci. 40 (2000) 1203-1206. doi:10.1021/ci000031d[Crossref]
[8] G.G. Cash, Permanental polynomials of smaller fullerenes, J. Chem. Inf. Comput. Sci. 40 (2000) 1207-1209. doi:10.1021/ci0000326[Crossref]
[9] R. Chen, A note on the relations between the permanental and characteristic polynomials of coronoid hydrocarbons, MATCH Commun. Math. Comput. Chem. 51 (2004) 137-148. | Zbl 1051.05058
[10] D. Cvetković, M. Doob and H. Sachs, Spectra of Graphs: Third Edition (Johann Ambrosius Barth Verlag, Heidelberg-Leipzig, 1995). | Zbl 0824.05046
[11] E.J. Farrell, An introduction to matching polynomials, J. Combin. Theory (B) 27 (1979) 75-86. doi:10.1016/0095-8956(79)90070-4[Crossref]
[12] I. Gutman and G.G. Cash, Relations between the permanental and characteristic polynomials of fullerenes and benzenoid hydrocarbons, MATCH Commun. Math. Comput. Chem. 45 (2002) 55-70. | Zbl 1026.05081
[13] D. Kasum, N. Trinajstić and I. Gutman, Chemical graph theory. III. On permanental polynomial , Croat. Chem. Acta 54 (1981) 321-328.
[14] L. Lovász and M.D. Plummer, Matching Theory, Annals of Discrete Mathematics, Vol. 29 (North-Holland, New York, 1986).
[15] R. Merris, K.R. Rebman and W. Watkins, Permanental polynomials of graphs, Linear Algebra Appl. 38 (1981) 273-288. doi:10.1016/0024-3795(81)90026-4[Crossref][WoS] | Zbl 0474.05049
[16] J. Turner, Generalized matrix functions and the graph isomorphism problem, SIAM J. Appl. Math. 16 (1968) 520-526. doi:10.1137/0116041[Crossref] | Zbl 0185.51701
[17] L.G. Valiant, The complexity of computing the permanent , Theoret. Comput. Sci. 8 (1979) 189-201. doi:10.1016/0304-3975(79)90044-6[Crossref]
[18] W. Yan and F. Zhang, On the permanental polynomials of some graphs, J. Math. Chem. 35 (2004) 175-188. doi:10.1023/B:JOMC.0000033254.54822.f8[Crossref] | Zbl 1048.05062
[19] H. Zhang and W. Li, Computing the permanental polynomials of bipartite graphs by Pfaffian orientation, Discrete Appl. Math. 160 (2012) 2069-2074. doi:10.1016/j.dam.2012.04.007 [Crossref][WoS] | Zbl 1246.05100