On γ-labelings of trees

Let G be a graph of order n and size m. A γ-labeling of G is a one-to-one function f:V(G) → {0,1,2,...,m} that induces a labeling f': E(G) → {1,2,...,m} of the edges of G defined by f'(e) = |f(u)-f(v)| for each edge e = uv of G. The value of a γ-labeling f is ${\displaystyle val\left(f\right)={\Sigma }_{e\in E\left(G\right)}f\prime K\left(e\right)}$. The maximum value of a γ-labeling of G is defined as ${\displaystyle va{l}_{max}\left(G\right)=max\{val\left(f\right):fisa\gamma -label\in gofG\}}$; while the minimum value of a γ-labeling of G is ${\displaystyle va{l}_{min}\left(G\right)=min\{val\left(f\right):fisa\gamma -label\in gofG\}}$; The values ${\displaystyle va{l}_{max}\left(S_{p,q}\right)}$ and ${\displaystyle va{l}_{min}\left(S_{p,q}\right)}$ are determined for double stars ${\displaystyle S_{p,q}}$. We present characterizations of connected graphs G of order n for which ${\displaystyle va{l}_{min}\left(G\right)=n}$ or ${\displaystyle va{l}_{min}\left(G\right)=n+1}$.