Eigensharp graphs: decomposition into complete bipartite subgraphs

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.