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 $val(f) = Σ_{e ∈ E(G)}f'K(e)$. The maximum value of a γ-labeling of G is defined as $val_{max}(G) = max {val(f) : f is a γ- labeling of G}$; while the minimum value of a γ-labeling of G is $val_{min}(G) = min {val(f) : f is a γ- labeling of G}$; The values $val_{max}(S_{p,q})$ and $val_{min}(S_{p,q})$ are determined for double stars $S_{p,q}$. We present characterizations of connected graphs G of order n for which $val_{min}(G) = n$ or $val_{min}(G) = n+1$