Spectral extremal problems on $[a,b]$-factors in binding graphs, and beyond

Abstract

The binding number $b(G)$ of a graph $G$ is the minimum value of $|N_G(X)|/|X|$ taken over all nonempty sets $X\subseteq V(G)$ such that $N_G(X)\ne V(G)$. A graph $G$ is 1-binding if $b(G)\ge 1$. Fan and Lin [Electron. J. Combin. 31 (2024) P1.30] characterized the unique spectral extremal graphs among 1-binding graphs with $\delta (G)\ge k$ that contain no $k$-factor for $k=1$, $2$. Subsequently, Tang and Zhang solved the problem for $k\ge 3$. Note that $\delta (G)\ge a$ is a necessary condition for the existence of an $[a,b]$-factor in a graph $G$. For $1\le a<b$, Fan et al. [Discrete Math. 349 (2026) 115103] determined the unique spectral extremal graph $H_n^{a,b}$ among connected graphs with $\delta (G)\ge a$ that contain no $[a,b]$-factor. Note that $b(H_n^{a,b})<1$. This motivates the corresponding spectral extremal problem for 1-binding graphs. Note that every 1-binding graph with $\delta (G)\ge 1$ contains a $[1,b]$-factor for $b>1$. For $2\le a<b$, we in this paper determine the unique extremal graph ${\mathcal{B}}_n^{a,b}$ with the maximum spectral radius among 1-binding graphs of order $n$ with $\delta (G)\ge a$ that contain no $[a,b]$-factor. The toughness of a non-complete graph $G$ is $t(G)=\min \{\frac{|X|}{c(G-X)}\mid X\subseteq V(G)\text{and}c(G-X)\ge 2\}$. A graph $G$ is 1-tough if $c(G-X)\le |X|$ for every $X\subseteq V(G)$ with $c(G-X)\ge 2$. Every 1-tough graph of order $n\ge 2$ is 1-binding. Note that $t({\mathcal{B}}_n^{a,b})<1$. This leads us to investigate the corresponding spectral extremal problem for 1-tough graphs. Note that every 1-tough graph with $\delta (G)\ge 1$ contains a $[1,b]$-factor for $b>1$. For $2\le a<b$, we in this paper characterize the unique extremal graph with the maximum spectral radius among 1-tough graphs of order $n$ with $\delta (G)\ge a$ containing no $[a,b]$-factor.

arXiv comment26 pages, 10 figures