Rank of signed cacti

Abstract

A signed cactus \(\dot{G}\) is a connected signed graph such that every edge belongs to at most one cycle. The rank of \(\dot{G}\) is the rank of its adjacency matrix. In this paper we prove that \[\sum_{i=1}^k n_i-2k\leq \operatorname{rank}(\dot{G})\leq \sum_{i=1}^k n_i-2t +2 s,\] where \(k\) is the number of cycles in \(\dot{G}\), \(n_1, n_2, \ldots, n_k\) are their lengths, \(t\) is the number of cycles whose rank is their order minus two, and \(s\) is the number of edges outside cycles. Signed cacti attaining the lower bound are determined.