Extremal Results for \(\mathcal{K}^-_{r + 1}\)-free Unbalanced Signed Graphs

Zhuang Xiong1, Yaoping Hou1
1College of Mathematics and Statistics, Hunan Normal University, small Changsha, Hunan 410081, China

Abstract

This paper investigates the Tur{\’a}n-like problem for \(\mathcal{K}^-_{r + 1}\)-free \((r \geq 2)\) unbalanced signed graphs, where \(\mathcal{K}^-_{r + 1}\) is the set of unbalanced signed complete graphs with \(r+1\) vertices. The maximum number of edges and the maximum index for \(\mathcal{K}^-_{r + 1}\)-free unbalanced signed graphs are given. Moreover, the extremal \(\mathcal{K}^-_{r + 1}\)-free unbalanced signed graphs with the maximum index are characterized.

Keywords: Signed graph, Extremal graph, Clique, Index