The index of unbalanced signed complete graphs whose negative-edge-induced subgraph is ${K}_{2,2}$-minor free

TL;DR

This paper characterizes the extremal signed complete graphs with a negative-edge-induced subgraph that is $K_{2,2}$-minor free, focusing on their maximum and second maximum spectral index.

Contribution

It provides a characterization of extremal signed complete graphs with a $K_{2,2}$-minor free negative-edge-induced subgraph, identifying those with the highest spectral indices.

Findings

Identifies extremal graphs with maximum index

Determines second maximum index configurations

Characterizes negative-edge-induced subgraphs that are $K_{2,2}$-minor free

Abstract

Let $\Gamma=(K_n,H^-)$ be a signed complete graph with the negative edges induced subgraph $H$. According to the properties of the negative-edge-induced subgraph, characterizing the extremum problem of the index of the signed complete graph is a concern in signed graphs. A graph $G$ is called $H$-minor free if $G$ has no minor which is isomorphic to $H$. In this paper, we characterize the extremal signed complete graphs that achieve the maximum and the second maximum index when $H$ is a $K_{2,2}$-minor free spanning subgraph of $K_n$.