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

DOI:
https://doi.org/10.1090/S0002-9947-1988-0929670-5

MathSciNet review:
929670

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be the minimum number of complete bipartite subgraphs needed to partition the edges of , and let be the larger of the number of positive and number of negative eigenvalues of . It is known that ; graphs with are called *eigensharp*. Eigensharp graphs include graphs, trees, cycles with or , prisms with , "twisted prisms" (also called "Möbius ladders") with or , 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 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.

**[1]**Dragoš M. Cvetković, Michael Doob, and Horst Sachs,*Spectra of graphs*, Pure and Applied Mathematics, vol. 87, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. Theory and application. MR**572262****[2]**Dragoš M. Cvetković and Ivan M. Gutman,*The algebraic multiplicity of the number zero in the spectrum of a bipartite graph*, Mat. Vesnik**9(24)**(1972), 141–150. MR**0323637****[3]**R. L. Graham and H. O. Pollak,*On embedding graphs in squashed cubes*, Graph theory and applications (Proc. Conf., Western Michigan Univ., Kalamazoo, Mich., 1972; dedicated to the memory of J. W. T. Youngs), Springer, Berlin, 1972, pp. 99–110. Lecture Notes in Math., Vol. 303. MR**0332576****[4]**Frank Harary, Derbiau Hsu, and Zevi Miller,*The biparticity of a graph*, J. Graph Theory**1**(1977), no. 2, 131–133. MR**0444523**, https://doi.org/10.1002/jgt.3190010208**[5]**A. J. Hoffman,*Eigenvalues and partitionings of the edges of a graph*, Linear Algebra and Appl.**5**(1972), 137–146. MR**0300937****[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 Tihany), Academic Press, 1969, pp. 231-236.**[7]**-, unpublished.**[8]**G. W. Peck,*A new proof of a theorem of Graham and Pollak*, Discrete Math.**49**(1984), no. 3, 327–328. MR**743808**, https://doi.org/10.1016/0012-365X(84)90174-2**[9]**Bruce Reznick, Prasoon Tiwari, and Douglas B. West,*Decomposition of product graphs into complete bipartite subgraphs*, Discrete Math.**57**(1985), no. 1-2, 189–193. MR**816059**, https://doi.org/10.1016/0012-365X(85)90167-0**[10]**H. Tverberg,*On the decomposition of 𝐾_{𝑛} into complete bipartite graphs*, J. Graph Theory**6**(1982), no. 4, 493–494. MR**679606**, https://doi.org/10.1002/jgt.3190060414

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1988-0929670-5

Keywords:
Decomposition,
bipartite subgraph,
graph product

Article copyright:
© Copyright 1988
American Mathematical Society