Toughness in regular graphs from eigenvalues

Ruifang Liu; Ao Fan; Jinlong Shu

Paper

Toughness in regular graphs from eigenvalues

Abstract

The {\it toughness}

for

which was initially proposed by Chvátal in 1973. A graph

is called {\it

-tough} if

Let

be the

-th largest eigenvalue of the adjacency matrix of a graph

. In 1996, Brouwer conjectured that

for a connected

-regular graph

where

Gu [SIAM J. Discrete Math. 35 (2021) 948-952] completely confirmed this conjecture. From Brouwer and Gu's result

we know that if

is a connected

-regular graph and

, then

for an integer

Inspired by the above result and utilizing typical spectral techniques and graph construction methods from Cioabă et al. [J. Combin. Theory Ser. B 99 (2009) 287-297], we prove that if

is a connected

-regular graph and

, then

. Meanwhile, we construct graphs implying that the upper bound on

is best possible. Our theorem strengthens the result of Chen et al. [Discrete Math. 348 (2025) 114404]. Finally, we also prove an upper bound of

to guarantee a connected

-regular graph to be

-tough.