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: 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.

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Keywords:
Decomposition,
bipartite subgraph,
graph product

Article copyright:
© Copyright 1988
American Mathematical Society