NONREGULAR GRAPHS WITH MINIMAL TOTAL IRREGULARITY

The total irregularity of a simple undirected graph G is defined as irrt(G) = 1/2 Sigma(u,v)is an element of V(G) vertical bar d(G)(u) - d(G)(v)vertical bar, where dG(u) denotes the degree of a vertex u is an element of V(G). Obviously, irr(t)(G) = 0 if and only if G is regular. Here, we characterise the nonregular graphs with minimal total irregularity and thereby resolve the recent conjecture by Zhu et al. ['The minimal total irregularity of graphs', Preprint, 2014, arXiv: 1404.0931v1] about the lower bound on the minimal total irregularity of nonregular connected graphs. We show that the conjectured lower bound of 2n-4 is attained only if nonregular connected graphs of even order are considered, while the sharp lower bound of n-1 is attained by graphs of odd order. We also characterise the nonregular graphs with the second and the third smallest total irregularity.