Primitive NonPowerful Symmetric LoopFree Signed Digraphs with Base 3 and Minimum Number of Arcs 
Received:December 27, 2011 Revised:September 03, 2012 
Key Word:
primitive symmetric nonpowerful 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). 

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Abstract: 
Let $S$ be a primitive nonpowerful symmetric loopfree signed digraph on even $n$ vertices with base 3 and minimum number of arcs. In [Lihua YOU, Yuhan WU. Primitive nonpowerful symmetric loopfree signed digraphs with given base and minimum number of arcs. Linear Algebra Appl., 2011, 434(5), 12151227], 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 $V=\{1,2,\ldots,n\}$, $A=\{(1,i),(i,1)\mid 3\leq i \leq n\} \cup \{(2i1,2i),(2i,2i1)\mid 2\leq i \leq \frac{n}{2}\}\cup \{(2,3),(3,2), (2,4),(4,2)\}$). 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. 
Citation: 
DOI:10.3770/j.issn:20952651.2013.03.002 
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