A study on parity signed graphs: The rna number

A signed graph (G,σ) on n vertices is called a parity if there bijection f:V(G)→{1,2,…,n} such that for each edge e=uv of G, f(u) and f(v) have the same σ(e)=1, opposite parities σ(e)=−1. The signature σ parity-signature G. Let Σ−(G) denote set number negative edges over all possible parity-signatures σ. rna σ−(G) G given by σ−(G)=minΣ−(G). In other words, minimum cardinality an edge-cut in two sides differ at most one. this paper, we prove any Σ−(G)={σ−(G)} only K1,n−1 with even or Kn. This confirms conjecture proposed Acharya Kureethara (2021)[1]. Moreover, nontrivial upper bounds number: m (n≥2) vertices, σ−(G)≤⌊m2(1+12⌈n2⌉−1)⌋≤⌊m2+n4⌋. We show Kn, Kn−e, Kn−▵ are graphs reaching bound ⌊m2+n4⌋. Finally, σ−(G)+σ−(G¯)≤σ−(G∪G¯), where G¯ complement solves problem et al. (2021)[2].