 Primitive Non-Powerful Symmetric Loop-Free Signed Digraphs with Base 3 and Minimum Number of Arcs
Received:December 27, 2011  Revised:September 03, 2012
Key Word: primitive   symmetric   non-powerful   base   signed digraph.
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant Nos.10901061; 11071088), Program on International Cooperation and Innovation, Department of Education, Guangdong Province (Grant No.2012gjhz0007) and the Zhujiang Technology New Star Foundation of Guangzhou City (Grant No.2011J2200090).
 Author Name Affiliation Lihua YOU School of Mathematical Sciences, South China Normal University, Guangdong 510631, P. R. China Yuhan WU School of Mathematical Sciences, South China Normal University, Guangdong 510631, P. R. China
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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.