Energy and Spectra of Zagreb Matrix of k-half Graph

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 k partite graph, a k-nested graph is defined. A half graph is a chain graph having no pairs of duplicate vertices. Similarly, a ’k-half graph’ is a class of k-nested graph with no pairs of duplicate vertices. The (first) Zagreb matrix or Z-matrix denoted by Z(G) = (z_ij)_n×n of a graph G, whose vertex v_i has degree d_i is defined by z_ij = d_i + d_j if the vertices v_i and v_j are adjacent and z_ij = 0 otherwise. Let ζ₁, ζ₂, …, ζ_n be the Zagreb eigenvalues of Z(G) and the Zagreb energy is the sum of the absolute values of the Zagreb eigenvalues. We obtain the determinant, eigenvalues and inverse of a k-half graph with respect to the Z-matrix. Bounds for the Zagreb energy and spectral radius are discussed along with the main and non-main Zagreb eigenvalues of a k-half graph.