Abstract
Some new constructions for families of cospectral graphs are derived, and some old ones are considerably generalized. One of our new constructions is sufficiently powerful to produce an estimated 72% of the 51039 graphs on 9 vertices which do not have unique spectrum. In fact, the number of graphs of ordern without unique spectrum is believed to be at leastαn 3 g −1 for someα>0, whereg n is the number of graphs of ordern andn ≥ 7.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Babai, L.,Almost all Steiner triple systems are asymmetric. Ann. Discrete Math.7 (1980), 37–39.
Bussemaker, F. C., Cobeljic, S., Cvetković, D. M. andSeidel, J. J.,Computer investigation of cubic graphs. Technishe Hogeschool Eindhoven, Report 76-WSK-01 (1976).
Cvetković, D. M.,The generating function for variations with restrictions and paths of the graph and self-complementary graphs. Univ. Beograd Publ. Elektrotehn. Fak., Ser. Mat. Fiz320–328 (1970), 27–34.
Cvetković, D. M., Doob, M. andSachs, H.,Spectra of graphs. Academic Press, New York, 1980.
Cvetković, D. M. andLučić, R. P.,A new generalization of the concept of the p-sum of graphs. Univ. Beograd Publ. Elektotehn. Fak., Ser. Mat. Fiz.302–319 (1970), 67–71.
Davidson, R. A.,Genesis and synthesis of cospectral and paracospectral graphs revelation of latest symmetries. Preprint, 1980.
Gantmakher, F. R.,Theory of matrices. Vols. 1, 2. Chelsea, New York, 1960.
Godsil, C. D.,Neighbourhoods of transitive graphs and GRRs. J. Comb. Theory, Ser. B29 (1980), 116–140.
Godsil, C. D. andMcKay, B. D.,Products of graphs and their spectra. Combinatorial Mathematics IV, Lecture Notes in Math.560, Springer-Verlag (1976), 61–72.
Godsil, C. D. andMcKay, B. D.,Some computational results on the spectra of graphs. Combinatorial Mathematics IV, Lecture Notes in Math.560, Springer-Verlag (1976), 73–92.
Lancaster, P.,Theory of matrices. Academic Press, New York, 1969.
McKay, B. D.,On the spectral characterisation of trees. Ars Combin.3 (1977), 219–232.
Seidel, J. J.,Graphs and two-graphs. InProc. Fifth Southeastern Conference on Combinatorics, Graph Theory and Computing (Boca Raton, FL., 1974), Congressus Numerantium X, Utilitas Math., Winnipeg, Man., 1974, pp. 125–143.
Schwenk, A. J.,Almost all trees are cospectral. InNew Directions in the Theory of Graphs (Proc. Third Ann Arbor Conf., Univ. Michigan, Ann Arbor, Michigan, 1971), Academic Press, New York, 1973, pp. 275–307.
Schwenk, A. J.,Removal-cospectral sets of vertices in a graph. InProc. Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Boca Raton, FL., 1979), Congressus Numerantium XXIII–XXIV, Utilitas Math., Winnipeg, Man., 1979, pp. 849–860.
Schwenk, A. J., Herndon, W. C. andEllzey, M. L.,The construction of cospectral composite graphs. InProc. Second Intern. Conf. on Combinatorial Math. (New York, 1978), Ann. N. Y. Acad. Science319 (1979), 490–496.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Godsil, C.D., McKay, B.D. Constructing cospectral graphs. Aeq. Math. 25, 257–268 (1982). https://doi.org/10.1007/BF02189621
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02189621