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 CTₙ (F) of the ring of all upper triangular matrices Tₙ(F) over a finite field F. We show that CTₙ (F) 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=pkn(n-1)/2 and |F|=pᵏ. In particular, if |F|=2, then the graph CTₙ (F) has 2ⁿ⁻¹ connected components, each component is isomorphic to the complete bipartite graph Km,m, where m=2n(n-1)/2. We also compute the diameter, triameter, and clique number of the graph CTₙ (F).