Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Eigensharp graphs: decomposition into complete bipartite subgraphs

Authors: Thomas Kratzke, Bruce Reznick and Douglas West
Journal: Trans. Amer. Math. Soc. 308 (1988), 637-653
MSC: Primary 05C50
MathSciNet review: 929670
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \tau (G)$ be the minimum number of complete bipartite subgraphs needed to partition the edges of $ G$, and let $ r(G)$ be the larger of the number of positive and number of negative eigenvalues of $ G$. It is known that $ \tau (G) \geqslant r(G)$; graphs with $ \tau (G) = r(G)$ are called eigensharp. Eigensharp graphs include graphs, trees, cycles $ {C_n}$ with $ n = 4$ or $ n \ne 4k$, prisms $ {C_n}\square {K_2}$ with $ n \ne 3k$, "twisted prisms" (also called "Möbius ladders") $ {M_n}$ with $ n = 3$ or $ n \ne 3k$, and some Cartesian products of cycles.

Under some conditions, the weak (Kronecker) product of eigensharp graphs is eigensharp. For example, the class of eigensharp graphs with the same number of positive and negative eigenvalues is closed under weak products. If each graph in a finite weak product is eigensharp, has no zero eigenvalues, and has a decomposition into $ \tau (G)$ stars, then the product is eigensharp. The hypotheses in this last result can be weakened. Finally, not all weak products of eigensharp graphs are eigensharp.

References [Enhancements On Off] (What's this?)

  • [1] D. M. Cvetković, M. Doob, and H. Sachs, Spectra of graphs, Academic Press, New York, 1979. MR 572262 (81i:05054)
  • [2] D. M. Cvetković and I. Gutman, The algebraic multiplicity of the number zero in the spectrum of a bipartite graph, Mat. Vesnik 9 (1972), 141-150. MR 0323637 (48:1993)
  • [3] R. L. Graham and H. O. Pollak, On embedding graphs in squashed cubes, Graph Theory and Applications (Proc. 2nd Internat. Conf. Graph Theory, Kalamazoo 1972), Lecture Notes in Math., vol. 303, Springer-Verlag, 1973, pp. 99-110. MR 0332576 (48:10902)
  • [4] F. Harary, D. F. Hsu, and Z. Miller, The biparticity of a graph, J. Graph Theory 1 (1977), 131-133. MR 0444523 (56:2874)
  • [5] A. J. Hoffman, Eigenvalues and partitionings of the edges of a graph, Linear Algebra Appl. 5 (1972), 137-146. MR 0300937 (46:97)
  • [6] L. Lovász, On coverings of graphs, Theory of Graphs (Proc. Conf. Tihany), Academic Press, 1969, pp. 231-236.
  • [7] -, unpublished.
  • [8] G. W. Peck, A new proof of a theorem of Graham and Pollak, Discrete Math. 49 (1984), 327-328. MR 743808 (85i:05201)
  • [9] B. Reznick, P. Tiwari, and D. B. West, Decomposition of product graphs into complete bipartite subgraphs, Discrete Math. 57 (1985), 179-183. MR 816059 (87i:05157)
  • [10] H. Tverberg, On the decomposition of $ {K_n}$ into complete bipartite subgraphs, J. Graph Theory 6 (1982), 493-494. MR 679606 (83m:05111)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 05C50

Retrieve articles in all journals with MSC: 05C50

Additional Information

Keywords: Decomposition, bipartite subgraph, graph product
Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society