Turán problem of signed graph for negative odd cycle

Abstract

We investigate natural Turán problems on signed graphs in this paper. Let $C_{2k+1}^{-}$ denote the signed cycle of length $2k+1$ with one negative edge. We determine the maximum number of edges among all unbalanced signed graphs of order $n$ with no subgraph switching to $C_{2k+1}^{-}$, where $3\le k\le \frac{n}{10}-1$, and characterize the extremal signed graphs. As a by-product, we also obtain the maximum spectral radius among these signed graphs.