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 \(t \geq l\). 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 \(n \geq 3\), we show \(L(n) = \{2, 3, \ldots, 2n\}\). Further, we characterize all primitive non-powerful signed symmetric digraphs of order \(n\) with at least one loop whose bases attain \(l(n)\).