Tur\'{a}n problem for $C_{2k+1}^{-}$-free signed graph

TL;DR

This paper investigates the maximum number of edges and the largest eigenvalue in unbalanced signed graphs that do not contain a negative cycle of odd length, establishing bounds and characterizing extremal graphs.

Contribution

It provides the first Turán-type bounds for $C_{2k+1}^{-}$-free unbalanced signed graphs and characterizes the extremal structures achieving these bounds.

Findings

Maximum edges in $C_{2k+1}^{-}$-free graphs are bounded by a specific extremal graph.

Largest eigenvalue is also bounded by that of the extremal graph.

Equality holds only for graphs switching equivalent to the extremal configuration.

Abstract

In this paper, we study the Tur\'{a}n problem for $C_{2k+1}^{-}$. Suppose that $\dot{G}$ is an unbalanced signed graph of order $n$ with $e(\dot{G})$ edges. Let $\lambda_{1} (\dot{G})$ be the largest eigenvalue of $\dot{G}$, and $C_{2k+1}^{-}$ be the set of the negative cycle with length $2k+1$($3 \le k \le \frac{n}{15}$). We prove that if $\dot{G}$ is a $C_{2k+1}^{-}$-free unbalanced signed graph, then $e(\dot{G}) \le e(C_{3}^{-} \cdot K_{n-2})$ and $\lambda_{1}(\dot{G}) \le \lambda_{1}(C_{3}^{-} \cdot K_{n-2})$, with equality holding if and only if $\dot{G}$ is switching equivalent to $C_{3}^{-} \cdot K_{n-2}$.