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The minimum number of complete bipartite subgraphs needed to partition the edges of a graph $G$ is denoted by $b(G)$. A known lower bound on $b(G)$ states that $b(G) \geq$ max $\{p(G), q(G)\}$, where $p(G)$ and $q(G)$ are the numbers of positive and negative eigenvalues of the adjacency matrix of $G$, respectively. When equality is attained, $G$ is said to be eigensharp and when $b(G)$=max$\{p(G), q(G)\} + 1$, $G$ is called an almost eigensharp graph. In this paper we investigate the eigensharpness of graphs with at most one cycle and products of some families of graphs. Among the other results, we show that $P_{m} \vee P_{n}, C_{m} \vee P_{n}$ for $m \equiv 2,3$ (mod 4) and $Q_{n}$ when $n$ is odd are eigensharp. We obtain some results on almost eigensharp graphs as well.
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