On the Second Zagreb Matrix of k-half Graphs

A bipartite graph in which the neighborhood of the vertices in each partite set forms a chain with respect to set inclusion is known as a chain graph. Recently, extending the concept of nesting from a bipartite graph to a k partite graph, a k-nested graph is defined. A chain graph without any pairs of duplicate vertices is a half graph. Similarly, a ’k-half graph’ is a class of k-nested graph with no pairs of duplicate vertices. The second Zagreb matrix or Z⁽²⁾-matrix denoted by Z⁽²⁾ (G) = (z_ij)_n×n of a graph G, whose vertex v_i has degree d_i is defined by z_ij = d_id_j if the vertices v_i and v_j are adjacent and z_ij = 0 otherwise. Suppose ζ⁽²⁾ 1^{, ζ(2)}2^{, …, ζ(2)}n are the eigenvalues of Z² (G), then the sum of the absolute values of the eigenvalues of Z⁽²⁾ (G) is called the second Zagreb energy of G. We obtain the determinant, eigenvalues and inverse of a k-half graph with respect to Z⁽²⁾ (G). Bounds for the second Zagreb energy and the spectral radius are discussed in this article, along with the main and non-main eigenvalues of a k-half graph with respect to Z⁽²⁾ (G).