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\begin{document}$$\varGamma $$\end{document} be a distance-semiregular graph on Y, and let DY\documentclass[12pt]{minimal}
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\begin{document}$$D^Y$$\end{document} be the diameter of Γ\documentclass[12pt]{minimal}
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\begin{document}$$\varGamma $$\end{document} on Y. Let Δ\documentclass[12pt]{minimal}
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\begin{document}$$\varDelta $$\end{document} be the halved graph of Γ\documentclass[12pt]{minimal}
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\begin{document}$$\varGamma $$\end{document} on Y. Fix x∈Y\documentclass[12pt]{minimal}
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\begin{document}$$x \in Y$$\end{document}. Let T and T′\documentclass[12pt]{minimal}
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\begin{document}$$T'$$\end{document} be the Terwilliger algebras of Γ\documentclass[12pt]{minimal}
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\begin{document}$$\varGamma $$\end{document} and Δ\documentclass[12pt]{minimal}
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\begin{document}$$\varDelta $$\end{document} with respect to x, respectively. Assume, for an integer i with 1≤2i≤DY\documentclass[12pt]{minimal}
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\begin{document}$$1 \le 2i \le D^Y$$\end{document} and for y,z∈Γ2i(x)\documentclass[12pt]{minimal}
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\begin{document}$$y,z \in \varGamma _{2i}(x)$$\end{document} with ∂Γ(y,z)=2\documentclass[12pt]{minimal}
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\begin{document}$$\partial _{\varGamma }(y,z)=2$$\end{document}, the numbers |Γ2i-1(x)∩Γ(y)∩Γ(z)|\documentclass[12pt]{minimal}
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\begin{document}$$|\varGamma _{2i-1}(x) \cap \varGamma (y) \cap \varGamma (z)|$$\end{document} and |Γ2i+1(x)∩Γ(y)∩Γ(z)|\documentclass[12pt]{minimal}
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\begin{document}$$|\varGamma _{2i+1}(x) \cap \varGamma (y) \cap \varGamma (z)|$$\end{document} depend only on i and do not depend on the choice of y, z. The first goal in this paper is to show the relations between T-modules of Γ\documentclass[12pt]{minimal}
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\begin{document}$$\varGamma $$\end{document} and T′\documentclass[12pt]{minimal}
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\begin{document}$$T'$$\end{document}-modules of Δ\documentclass[12pt]{minimal}
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\begin{document}$$\varDelta $$\end{document}. Assume Γ\documentclass[12pt]{minimal}
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\begin{document}$$\varGamma $$\end{document} is the incidence graph of the Hamming graph H(D, n) on the vertex set Y and the set C\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {C}}$$\end{document} of all maximal cliques. Then, Γ\documentclass[12pt]{minimal}
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\begin{document}$$\varGamma $$\end{document} satisfies above assumption and Δ\documentclass[12pt]{minimal}
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\begin{document}$$\varDelta $$\end{document} is isomorphic to H(D, n). The second goal is to determine the irreducible T-modules of Γ\documentclass[12pt]{minimal}
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\begin{document}$$\varGamma $$\end{document}. For each irreducible T-module W, we give a basis for W the action of the adjacency matrix on this basis and we calculate the multiplicity of W.
