Bulletin of the
Korean Mathematical Society BKMS

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) > 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 and almost eigensharpness of lexicographic product of some graphs.

Keywords: Lexicographic product of graphs, Biclique partition number, Eigensharp graphs.

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