On the Irregularity Cost of a Tree

A multigraph is irregular if no two of its vertices have the same degree. It is known that every graph \(G\) with at most one trivial component and no component isomorphic to \(K_2\) is the underlying graph of some irregular multigraph. The irregularity cost of a graph with at most one trivial component and no component isomorphic to \(K_2\) is defined by \(ic(G) = \min\{|{E}(H)| – |{E}(G)| \mid H \text{ is an irregular multigraph containing } G \text{ as underlying graph}\}\). It is shown that if \(T\) is a tree on \(n\) vertices, then

\[\frac{n^2-3n+4}{4}\quad \leq \quad ic(T) \leq \binom{n-1}{2}\: \text{if}\: n\equiv0 \;\text{or}\; 3\pmod{4} \; \text{and}\]
\[\frac{n^2-3n+6}{4}\quad \leq \quad ic(T) \leq \binom{n-1}{2}\: \text{if}\: n\equiv1 \;\text{or}\; 2\pmod4 \]
Furthermore, these bounds are shown to be sharp.