The unitary Cayley graph of upper triangular matrix rings

The unitary Cayley graph CR of a finite unital ring R is the simple graph with vertex set R in which two elements x and y are connected by an edge if and only if x-y is a unit of R. We characterize the unitary Cayley graph Cₓ䂸 (₅) of the ring of all upper triangular matrices Tₙ (F) over a finite field F. We show that Cₓ䂸 (₅) is isomorphic to the semistrong product of the complete graph Kₘ and the antipodal graph of the Hamming graph A (H (n, pᵏ) ), where m=p^kn (n-1) {2} and |F|=pᵏ. In particular, if |F|=2, then the graph Cₓ䂸 (₅) has 2^n-1 connected components, each component is isomorphic to the complete bipartite graph K₌, ₌, where m=2^n (n-1) {2}. We also compute the diameter, triameter, and clique number of the graph Cₓ䂸 (₅).