The total graphs of finite rings

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

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