Standard Module And Imaginary Adjacency Matrix Of Hamming Graph

In this thesis,by using the theories of complete graphs and cartesian products of graphs,we study the following properties of Hamming graph H（D,n）.Firstly,we construct a basis {（?）|y ∈ X} for the standard module V of H（D,n）and discuss its properties concerning with the Hadamard multiplication on V.For 0 ≤ i≤ D,we show that {（?）|y E ri（x）is a basis for EiV,where Ei（0 ≤ i ≤ D）is the primitive idempotent of H（,n）,and study the properties of {（?）|y∈Γi（x）} concerning with the Norton multiplication on E₁V.Secondly,we show that there exists an imaginary adjacency matrix A for H（D,n）with respect to a fix vertex x.Let A denote the adjacency matrix of H（D,n）and let A*denote the dual adjacency matrix of H（D,n）with respect to x.We construct an automorphism P of the Terwilliger algebra of H（D,n）such that P（A）A*＝A*,P（A*）= Aε,P（Aε）＝ A.We construct three anti-automorphisms σ₁,σ₂,σ₃ of the Terwilliger algebra of H（D,n）such that σ₁（A）= A,σ₁（A*）= Aε＝ σ₁（Aε）= A*;σ₂（A*）= A*,σ₂（A）= Aε,σ₂（Aε）= A;σ₃（Aε）= Aε,σ₃（A）= A*,σ₃（A*）= A.