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

TL;DR

This work addresses the spectral Turán problem for unbalanced signed complete graphs with the negative-edge-induced subgraph constrained to be $K_{2,2}$-minor free. It develops and applies equitable partition and quotient-matrix techniques, together with switching and relocation arguments, to identify the extremal structures that maximize the adjacency-spectral radius $\,\lambda_1(A(\Gamma))\$ among all such signed graphs. The main finding is that the maximum is achieved by $\Gamma=(K_n,U_1^-)$ and the second maximum by $\Gamma=(K_n,D_{1,n-3}^-)$ (up to switching isomorphism) for $n\ge 5$, with a complete, case-based proof that rules out all other admissible $H$. The results contribute to the broader understanding of spectral extremal problems in signed graphs and illustrate how specific minor-free constraints on the negative-edge subgraph shape the extremal frequency and structure of edge signs in maximizing configurations.

Abstract

Let $Γ=(K_n,H^-)$ 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 $K_n$.

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Theorems & Definitions (24)

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