The unitary Cayley graph $C_R$ 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 $C_{ T_n (\mathbb{ F})}$ of the ring of all upper triangular matrices $T_n(\mathbb{ F})$ over a finite field $\mathbb{ F}$. We show that $C_{ T_n (\mathbb{ F})}$ is isomorphic to the semistrong product of the complete graph $K_m$ and the antipodal graph of the Hamming graph $A(H(n,p^k))$, where $m=p^{ \frac{ kn(n-1)}{ 2}}$ and $|\mathbb{ F}|=p^k$. In particular, if $|\mathbb{ F}|=2$, then the graph $C_{ T_n (\mathbb{ F})}$ has $2^{ n-1}$ connected components, each component is isomorphic to the complete bipartite graph $K_{ m,m}$, where $m=2^{ \frac{ n(n-1)}{ 2}}$. We also compute the diameter, triameter, and clique number of the graph $C_{ T_n (\mathbb{ F})}$.
Categories: math.CO math.RA