On the spectrum and main eigenvalues of k-half graphs

A chain graph is a bipartite graph in which the neighborhood of the vertices in each partite set forms a chain with respect to set inclusion. By extending the concept of nesting from a bipartite graph to a kpartite graph, a k-nested graph is defined. The half graph is a chain graph with no pairs of duplicate vertices. Similarly, the ’k-half graph’ is a class of k-nested graph without any duplicate vertices. We study some spectral properties of a k-half graph. We prove that a k-half graph on kn vertices has exactly n main eigenvalues, and there are 2k downer vertices with respect to each eigenvalue of its adjacency matrix. We show the existence of [Formula] edges in a k-half graph on kn vertices, which are 2-downer for a few eigenvalues.