Let $K$ be any finite field. For any prime $p$, the $p$-adic valuation map is given
by $\psi_{p}:K/\{0\} \to \R^+\bigcup\{0\}$ is given by $\psi_{p}(r) = n$ where $r =
p^n \frac{a}{b}$, where $p,a,b$ are relatively prime. The field $K$ together with a
valuation is called valued field. Also, any field $K$ has the trivial valuation
determined by $\psi{(K)} = \{0,1\}$. Through out the paper K represents $\Z_q$. In this paper, we construct the graph
corresponding to the valuation map called the valued field graph, denoted by
$VFG_{p}(\Z_{q})$ whose vertex set is $\{v_0,v_1,v_2,\ldots, v_{q-1}\}$ where two
vertices $v_i$ and $v_j$ are adjacent if $\psi_{p}(i) = j$ or $\psi_{p}(j) = i$.
Here, we tried to characterize the valued field graph in $\Z_q$. Also we analyse various graph theoretical parameters such as diameter, independence number etc.
Counting the number of distinct distances of elements in valued field extensions
