Primitive Non-Powerful Symmetric Loop-Free Signed Digraphs with Base 3 and Minimum Number of Arcs

Abstract:

      Let $S$ be a primitive non-powerful symmetric loop-free signed digraph on even $n$ vertices with base 3 and minimum number of arcs. In [Lihua YOU, Yuhan WU. Primitive non-powerful symmetric loop-free signed digraphs with given base and minimum number of arcs. Linear Algebra Appl., 2011, 434(5), 1215--1227], authors conjectured that $D$ is the underlying digraph of $S$ with $\exp(D)=3$ if and only if $D$ is isomorphic to $ED_{n,3,3}$, where $ED_{n,3,3}=(V,A)$ is a digraph with $V=\{1,2,\ldots,n\}$, $A=\{(1,i),(i,1)\mid 3\leq i \leq n\} \cup \{(2i-1,2i),(2i,2i-1)\mid 2\leq i \leq \frac{n}{2}\}\cup \{(2,3),(3,2), (2,4),(4,2)\}$). In this paper, we show the conjecture is true and completely characterize the underlying digraphs which have base 3 and the minimum number of arcs.