Non-regular graphs with minimal total irregularity

The total irregularity of a simple undirected graphG is defined as irrt(G) = 1 2 ∑ u,v∈V (G) |dG(u)− dG(v)|, where dG(u) denotes the degree of a vertex u ∈ V (G). Obviously, irrt(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 [18] 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.