Rank of signed cacti

Abstract

A signed cactus G˙ is a connected signed graph such that every edge belongs to at most one cycle. The rank of G˙ is the rank of its adjacency matrix. In this paper we prove that $$\sum\limits_{ i=1}^n n_i -2k \leq rank(G') \leq \sum\limits_{i=1}^n n_i- 2t+2s$$ where $k$ is the the number of cycles in $G'$, $n_1,n_2, \ldots n_k$ are their lengths, $t$ is number of cycles which rank is their order minus two, and $s$ is the number of edges outside the cycles. Signed cacti attaining the lower bound are determined.