Bases of Primitive Non-Powerful Signed Symmetric Digraphs with Loops

Abstract

Let $S$ be a primitive non-powerful signed digraph. The base $l(S)$ of $S$ is the smallest positive integer $l$ such that for all ordered pairs of vertices $i$ and $j$ (not necessarily distinct), there exists a pair of $SSSD$ walks of length $t$ from $i$ to $j$ for each integer $t\ge l$. In this work, we use $PNSSD$ to denote the class of all primitive non-powerful signed symmetric digraphs of order $n$ with at least one loop. Let $l(n)$ be the largest value of $l(S)$ for $S\in$ $PNSSD$, and $L(n)=\{l(S)|S\in PNSSD\}$. For $n\ge 3$, we show $L(n)=\{2,3,\dots ,2n\}$. Further, we characterize all primitive non-powerful signed symmetric digraphs of order $n$ with at least one loop whose bases attain $l(n)$.