  The {\it total irregularity} of a simple undirected graph $G$ is defined as
${\rm irr}_t(G) =$ $\frac{1}{2}\sum_{u,v \in V(G)}$ $\left| d_G(u)-d_G(v)
\right|$, where $d_G(u)$ denotes the degree of a vertex $u \in V(G)$.
Obviously, ${\rm irr}_t(G)=0$ if and only if $G$ is regular. Here, we
characterize the non-regular graphs with minimal total irregularity and thereby
resolve the recent conjecture by Zhu, You and Yang~\cite{zyy-mtig-2014} about
the lower bound on the minimal total irregularity of non-regular connected
graphs. We show that the conjectured lower bound of $2n-4$ is attained only if
non-regular connected graphs of even order are considered, while the sharp
lower bound of $n-1$ is attained by graphs of odd order. We also characterize
the non-regular graphs with the second and the third smallest total
irregularity.
