Abstract
We examined trees with one multiple edge (of multiplicityk) and report all isospectral graphs found when the number of vertices was n < 9. Tile search for isospectrul multitrees was carried out systematically by constructing the characteristic polynomials of all trees having one weighted edge. For all multitrees havingn < 7 vertices, we tabulated the coefficients of the characteristic polynomial. We restricted the analysis to trees with Me maximal valencyd = 4. The number of graphs considered exceeds 300. The smallest pair of isospectral multitrees (i.e. trees with a multiple edge) hasn = 6 vertices, There is a pair of trees whenn = 7, three pairs whenn = 8, and five pads whenn = 9. In all cases, whenk = I is assumed, isospectral multitrees reduce to the same tree. Whenk = 0 is assume(], isospectral trees produce either the same disconnected graph, or an isospectral forest.
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References
R.J. Wilson,Introduction to Graph Theory (Academic, New York, 1972).
M. Randić, N. Trinajstić and T. Živković, J.C.S. Faraday Trans. II, 72 (1976)244.
A.J. Schwenk, in:New Directions in the Theory of Graphs, ed. F. Harary (Academic, New York, 1973) pp. 275–307.
A.J. Schwenk, New York Acad. Sci. (1973)275.
F. Harary, C. King, A. Mowshowitz and R.C. Read, Bull. London Math. Soc. 3 (1971)321.
N. Trinajstić, Croat. Chem. Acta 49 (1977)593; S.S. Arnato, B.M. Gimarc and N. Trinajstić, Croat. Chem. Acta 54(1981)1.
L. Spialter, J. Chem. Docum. 4 (1964)269.
H. Sachs, Publ. Math. (Debrecen) 11 (1964)119.
C.A. Coulson, Proc. Camb. Phil. Soc. 46 (1950)202.
D. König,Theorie endlichen and unendlichen Graphen (1936).
A.N. Krylow, Izv. Akad. Nauk SSSR, Ser. 7, Fiz-Mat. 4 (1931)491 (in Russian).
U.J.J. Le Verrier, J. Math. 5 (1840)95, 220.
H. Weyland, Quart. Appl. Math. 2 (1945)222.
K. Balasubramanian, Theor. Chim. Acta 65 (1984)49; P. Křivka, Z. Jeričević and N. Trinajstić, Int. J. Quant. Chem: Quant. Chem. Symp. 19 (1980129. R. Barakat, Theor. Chim. Acta 69(1986)35; J. Brocas, Theor. Chim. Acta 68(1985)155.
M. Randić, SIAM J. Alg. Disc. Meth. 6 (1985)145; I. Gutman and A. Shalabi, MATCH 18(1985)3.
K. Balasubramanian, J. Comput. Chem. 5 (1984)387.
MATLAB is an interactive computer program that serves as a convenient “laboratory” for computations involving matrices (written in FORTRAN). For more information, consult: C. Moles, MATLABBeginners User's Guide, Department of Computer Sci., University of New Mexico (1982).
K. Balasubramanian, Int. J. Quant. Chem. 22 (1982)581.
K. Balasubramanhn and M. Randić, Theor. Chim. Acta 61 (1982)307; K. Balasubramanian and M. Randić, Int. J. Quant. Chem. 28(1985)481.
M. Randić, J. Comput. Chem. 3 (1982)421; M. Randić, Theor. Chim. Acta 62(1983)485; H. Hosoya and M. Randić, Theor. Chim. Acta 63(1983)473.
M. Randić, B. Baker and A.F. Kleiner, Int. J. Quant. Chem: Quant. Chem. Symp. 19 (1986)107.
M. Randić and B. Baker, Tables of the characteristic polynomials for multitrees, Preprint, Drake University (1987).
For a discussion of the Catalan numbers relevant to chemistry see: N. Ohkami and H. Hosoya, Bull. Chem. Soc. Japan 52 (1979)1264.
M. Randić, H. Hosoya, N. Ohkami and N. Trinajstić, J. Math. Chem. 1 (1987)97.
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Dedicated to Basil E. Gillam, Professor emeritus of the Department of Mathematics and Computer Science at Drake University.
Reported in part at the 1986 International Congress of Mathematicians, Berkeley, California, USA.
Operated for the U.S. Department of Energy by the Iowa State University under Contract No. W-7405-Eng-82. This work was supported in part by the Office of R.S. Hansen, Director.
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Randic, M., Baker, B. Isorpectral multitrees. J Math Chem 2, 249–265 (1988). https://doi.org/10.1007/BF01167205
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DOI: https://doi.org/10.1007/BF01167205