Spectral Tur\'an problem for $t\mathcal{K}_{4}^{-}$-free unbalanced signed graphs

TL;DR

This paper characterizes the extremal unbalanced signed graphs, avoiding certain subgraphs, that maximize spectral radius, contributing to spectral graph theory and extremal combinatorics.

Contribution

It provides a complete characterization of extremal graphs for the spectral radius in the context of $t ext{K}_4^-$-free unbalanced signed graphs.

Findings

Identifies extremal graphs achieving maximum spectral radius.

Characterizes the structure of $t ext{K}_4^-$-free unbalanced signed graphs.

Advances understanding of spectral properties in signed graph extremal problems.

Abstract

Let $tK_4$ denote the family of all graphs consisting of $t$ copies of $K_4$ that are allowed to share vertices and $t\mathcal{K}_{4}^{-}$ be the set of all unbalanced signed graphs whose underlying graphs are elements of $tK_4$. In this paper, we characterize the extremal graphs that achieve the maximum index and spectral radius among all $t\mathcal{K}_{4}^{-}$-free unbalanced signed graphs with given order.