On the Characterization of the Unitary Cayley Graphs of the Upper Triangular Matrix Rings

There are several graphs naturally associated with rings. The unitary Cayley graph of a ring R is the graph with vertex set R, where two elements x,y∈R are adjacent if and only if x−y is a unit of R. We show that the unitary Cayley graph CTn(F) of the ring Tn(F) of all upper triangular matrices over a finite field F is isomorphic to a semistrong product of a complete graph and the antipodal graph of a Hamming graph. In particular, when |F|=2, the graph CTn(F) has a highly symmetric structure: it is the union of 2n−1 complete bipartite graphs. Moreover, we prove that the clique number and the chromatic number of CTn(F) are both equal to |F|, and we establish tight upper and lower bounds for the domination number of CTn(F).