Comparing the largest eigenvalue on adjacency and antiadjacency matrices of graphs which constructed using Boolean operation (⊕ and ⊕)

Let G be a graph with V(G) = {v₁, ., v_n} and E(G) = {e₁, ., e_m}. In this paper, only graphs with no multiple edges are observed. The n × n matrix A = [a_ij], where a_ij = 1 if e = v_i v_j ϵ E(G) and a_ij = 0 otherwise, is the adjacency matrix of G and is denoted by A(G). The n × n matrix B = [b_ij], where b_ij = 0 if e = v_i v_j ϵ E(G) and b_ij = 1 otherwise, is the antiadjacency matrix of G and is denoted by B(G). Boolean operations on two graphs have been examined by Harary and Wilcox. Hence, this paper will consider Boolean operations on two adjancency and antiadjancency matrices of two graphs G₁ and G₂ with V(G₁) = V(G₂). Boolean operations which are reviewed on this paper are OR (V), AND (λ), XOR(⊕), and NXOR(⊕) However, the paper only focus on operation ⊕ and ⊕. The purposes of this paper are to introduce the operations on two adjacency and antiadjacency matrices of two graphs G₁ and G₂ with V(G₁) = V(G₂), to reveal the effect on the represented graph using operations ⊕ and ⊕ on both adjacency and antiadjacency matrices, and to compare the largest eigenvalues between the matrices generated by the Boolean operations.