Induced subgraphs of product graphs and a generalization of Huang's theorem

TL;DR

This paper generalizes Huang's theorem on induced subgraphs with high maximum degree from hypercubes to Cartesian and semi-strong product graphs, using spectral properties of signed graphs.

Contribution

It introduces new signed product graphs and establishes bounds on maximum degree of induced subgraphs based on eigenvalues, extending Huang's result.

Findings

Induced subgraphs of product graphs have high maximum degree under certain spectral conditions.

Spectral symmetry conditions relate to the maximum degree bounds.

Provides necessary and sufficient conditions for spectral symmetry of the product graphs.

Abstract

Recently, Huang showed that every $(2^{n-1}+1)$-vertex induced subgraph of the $n$-dimensional hypercube has maximum degree at least $\sqrt{n}$ in [Annals of Mathematics, 190 (2019), 949--955]. In this paper, we discuss the induced subgraphs of Cartesian product graphs and semi-strong product graphs to generalize Huang's result. Let $\Gamma_1$ be a connected signed bipartite graph of order $n$ and $\Gamma_2$ be a connected signed graph of order $m$. By defining two kinds of signed product of $\Gamma_1$ and $\Gamma_2$, denoted by $\Gamma_1\widetilde{\Box}\Gamma_2$ and $\Gamma_1\widetilde{\bowtie} \Gamma_2$, we show that if $\Gamma_1$ and $\Gamma_2$ have exactly two distinct adjacency eigenvalues $\pm\theta_1$ and $\pm\theta_2$ respectively, then every $(\frac{1}{2}mn+1)$-vertex induced subgraph of $\Gamma_1\widetilde{\Box}\Gamma_2$ (resp. $\Gamma_1\widetilde{\bowtie} \Gamma_2$) has maximum degree at least $\sqrt{\theta_1^2+\theta_2^2}$ (resp. $\sqrt{(\theta_1^2+1)\theta_2^2}$). Moreover, we discuss the eigenvalues of $\Gamma_1\widetilde{\Box} \Gamma_2$ and $\Gamma_1\widetilde{\bowtie} \Gamma_2$ and obtain a sufficient and necessary condition such that the spectrum of $\Gamma_1\widetilde{\Box}\Gamma_2$ and $\Gamma_1\widetilde{\bowtie}\Gamma_2$ are symmetric, from which we obtain more general results on maximum degree of the induced subgraphs.