SCORE SETS IN ORIENTED GRAPHS

The score of a vertex v in an oriented graph $D$ is$a_{v}=n-1+d^{+}_{v}-d_{v}^{-}$, where $d^{+}_{v}$ and $d^{-}_{v}$ are the outdegree andindegree respectively of v and n is the number of vertices in $D.$ The set of distinct scoresof the vertices in an oriented graph $D$ is called its score set. If $a>0$ and $d>1$ arepositive integers, we show there exists an oriented graph with score set$lbrace a, ad, ad^{2},ldots,ad^{n}brace$ except for $a=1$, $d=2$, $n>0$, and for$a=1$, $d=3$, $n>0$. It is also shown that there exists no oriented graph with score set$lbrace a,ad,ad^{2},ldots,ad^{n}brace$, $n>0$ when either $a=1$, $d=2$, or $a=1$, $d=3$.Also we prove for the non-negative integers $a_{1},a_{2},ldots,a_{n}$ with$a_{1}1$}, a_{i},&mbox{for $i=1$}.end{array}ight.$$