Determinant of antiadjacency matrix of union and join operation from two disjoint of several classes of graphs

Let G be a graph with V(G) = {v₁, ..., v_n} and E(G) = {e₁, ...,e_m}. We only consider undirected graphs with no multiple edges in this paper. The adjacency matrix of G, denoted by A(G), is the n × n matrix A = [a_ij], where a_ij = 1 if e = v_iv_j ∈ E(G) or otherwise a_ij = 0. The anti adjacency matrix of G, denoted by B(G), is the n × n matrix B = [b_ij], where b_ij = 0 if e = v_iv_j ∈ E(G) or otherwise b_ij = 1. Properties of the determinant of the adjacency matrix of some simple graphs have been studied by many researchers. However, the determinant of the anti-adjacency matrix has not been explored yet. If G₁ and G₂ are disjoint graphs, then the joining of two graphs G₁ and G₂, denoted G₁ G₂ is defined by taking copies of G₁ and G₂ and adding edges so that each vertex in G₁ is adjacent to every vertex in G₂. In this paper, we show the properties of the determinant of joining two graphs, G₁ and G₂. Union of two graphs, denote G₁ ∪ G₂ is a graph formed by taking copies of G₁ and G₂. The objectives of this paper are to identify some properties of the determinant anti adjacency matrix of joining and union operation from two disjoint graphs. This paper also emphasizes on investigating the determinant of some special graph class formed by joining and unioning operation of two disjoint of several classes of graphs, such as Bipartite graphs, Cycles, Complete graphs, Stars, and Wheels.