The Optimization of Signed Trees

A signed graph $G$ is a graph where each edge is assigned a + (positive edge) or a - (negative edge). The signed degree of a vertex $v$ in a signed graph, denoted by $sdeg(v)$, is the number of positive edges incident to $v$ subtracted by the number of negative edges incident to $v$. Finally, we say $G$ realizes the set $D$ if: $$ D = \{sdeg(v) \text{ : } v\in V(G) \}. $$ The topic of signed degree sets and signed degree sequences has been studied from many directions. In this paper, we study properties needed for signed trees to have a given signed degree set. We start by proving that $D$ is the signed degree set of a tree if and only if $1\in D$ or $-1\in D$. Further, for every valid set $D$, we find the smallest diameter that a tree must have to realize $D$. Lastly, for valid sets $D$ with nonnegative numbers, we find the smallest order that a tree must have to realize $D$.