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A radio antipodal coloring of a connected graph $G$ with diameter $d$ is an assignment of positive integers to the vertices of $G$, with $x\in V\left(G\right)$ assigned $c\left(x\right)$, such that $$d(u,v)+\left|c\right(u)-c(v\left)\right|\ge d$$ for every two distinct vertices $u$, $v$ of $G$, where $d(u,v)$ is the distance between $u$ and $v$ in $G$. The radio antipodal coloring number $\mathrm{a}c\left(c\right)$ of a radio antipodal coloring $c$ of $G$ is the maximum color assigned to a vertex of $G$. The radio antipodal chromatic number $\mathrm{a}c\left(G\right)$ of $G$ is $min\left\{\mathrm{a}c\right(c\left)\right\}$ over all radio antipodal colorings $c$ of $G$. Radio antipodal chromatic numbers of paths...

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 $val\left(f\right)={\Sigma }_{e\in E\left(G\right)}{f}^{\text{'}}K\left(e\right)$. The maximum value of a γ-labeling of G is defined as $va{l}_{max}\left(G\right)=maxval\left(f\right):fisa\gamma -labelingofG$; while the minimum value of a γ-labeling of G is $va{l}_{min}\left(G\right)=minval\left(f\right):fisa\gamma -labelingofG$; The values $va{l}_{max}\left(S_{p,q}\right)$ and $va{l}_{min}\left(S_{p,q}\right)$ are determined for double stars $S_{p,q}$. We present characterizations of connected graphs G of order n for which $va{l}_{min}\left(G\right)=n$ or $va{l}_{min}\left(G\right)=n+1$.