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

Abstract

Let G be a graph with V(G) = {v1, …, vn} and E(G) = {e1, …, .em}. 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 = [aij], where aij = 1 if e = vivj ∈ E(G) or otherwise aij = 0. The anti adjacency matrix of G, denoted by B(G), is the n × n matrix B = [bij], where bij = 0 if e = vivj ∈ E(G) or otherwise bij = 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.