Bulletin of the
Korean Mathematical Society BKMS

The minimum number of complete bipartite subgraphs \linebreak needed to partition the edges of a graph $G$ is denoted by $b(G)$. A known lower bound on $b(G)$ states that $b(G)\geq \max\lbrace p(G), q(G)\rbrace$, 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 $b(G) =\max \lbrace p(G), q(G)\rbrace + 1$, $G$ is called an almost eigensharp graph. In this paper, we investigate the eigensharpness and almost eigensharpness of lexicographic products of some graphs.

Keywords: Lexicographic product of graphs, biclique partition number, eigensharp graphs

MSC numbers: Primary 05C76, 05C99, 15A18