The Local Bases of Primitive Non-Powerful Signed Symmetric Digraphs with Loops

Abstract

Let $S$ be a primitive non-powerful signed digraph of order $n$. The base of a vertex $u$, denoted by $l_S(u)$, is the smallest positive integer $l$ such that there is a pair of SSSD walks of length $i$ from $u$ to each vertex $v\in V(S)$ for any integer $t\ge l$. We choose to order the vertices of $S$ in such a way that $l_S(1)\le l_S(2)\le \dots \le l_S(n)$, and call $l_S(k)$ the $k$th local base of $S$ for $1\le k\le n$. 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 $l(k)$ be the largest value of $l_S(k)$ for $S\in$ PNSSD, and $L(k)=\{l_S(k)|S\in PNSSD\}$. For $n\ge 3$ and $1\le k\le n-1$, we show $I(k)=2n–1$ and $L(k)=\{2,3,\dots ,2n-1\}$. Further, we characterize all primitive non-powerful signed symmetric digraphs whose $k$th local bases attain $I(k)$.