Toughness and Aα-spectral radius in graphs

TL;DR

The paper links graph toughness to the $A_{α}$-spectral radius, defining the $A_{α}$-matrix $A_{α}(G)$ and the order threshold $f(α)$. It proves a tight 1-toughness condition: if $n≥f(α)$ and $ρ_{α}(G)≥ρ_{α}(K_1∨(K_{n-2}∪K_1))$, then $G$ is 1-tough unless it is the extremal graph $K_1∨(K_{n-2}∪K_1)$, with the threshold given by the largest root of a cubic polynomial in $x$. The second main result extends to $t$-tough graphs for $α∈[1/2,3/4)$, providing a similar spectral-condition bound for $n$ and identifying the extremal graph $K_{2t-1}∨(K_{n-2t}∪K_1)$. The proofs combine subgraph comparison, equitable partitions, and standard spectral bounds to derive a cubic-root threshold governing toughness via the $A_{α}$-spectral radius, offering practical criteria for verifying toughness from spectral data.

Abstract

Let $α\in[0,1)$, and let $G$ be a connected graph of order $n$ with $n\geq f(α)$, where $f(α)=6$ for $α\in[0,\frac{2}{3}]$ and $f(α)=\frac{4}{1-α}$ for $α\in(\frac{2}{3},1)$. A graph $G$ is said to be $t$-tough if $|S|\geq tc(G-S)$ for each subset $S$ of $V(G)$ with $c(G-S)\geq2$, where $c(G-S)$ is the number of connected components in $G-S$. The $A_α$-spectral radius of $G$ is denoted by $ρ_α(G)$. In this paper, it is verified that $G$ is a 1-tough graph unless $G=K_1\vee(K_{n-2}\cup K_1)$ if $ρ_α(G)\geqρ_α(K_1\vee(K_{n-2}\cup K_1))$, where $ρ_α(K_1\vee(K_{n-2}\cup K_1))$ equals the largest root of $x^{3}-((α+1)n+α-3)x^{2}+(αn^{2}+(α^{2}-α-1)n-2α+1)x-α^{2}n^{2}+(3α^{2}-α+1)n-4α^{2}+5α-3=0$. Further, we present an $A_α$-spectral radius condition for a graph to be a $t$-tough graph.