Abstract
A non-negative graph invariant IM is said to be an irregularity measure of a graph G if the following condition holds: IM(G)=0 if and only if G is regular. There exist many irregularity measures in the literature and the Albertson index is probably the most studied such measure. In order to overcome several drawbacks of the Albertson index, a variant of the Albertson index was recently introduced under the name ǣtotal irregularityǥ. The total irregularity index of a graph G is defined as 12∑u,w∈V(G)|degG(u)−degG(w)|, where degG(w) is the degree of a vertex w∈V(G). By an n-vertex tree, we mean a tree of order n. In the present study, the best possible sharp upper and lower bounds on the total irregularity index of n-vertex trees with fixed number of segments or branching vertices are derived.