The total graphs of finite rings

In this paper we extend the study of total graphs $\tau (R)$ to non-commutative finite rings $R$. We prove that $\tau (R)$ is connected if and only if $R$ is not local and we see that in that case $\tau (R)$ is always Hamiltonian. We also find an upper bound for the domination number of $\tau (R)$ for all finite rings $R$.