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Let G be a connected graph. The toughness of G is defined ast(G)=min{\frac{|S|}{c(G-S)}}, in which the minimum is taken over all propersubsets S\subset V(G) such that c(G-S)\geq 2 where c(G-S) denotes the number ofcomponents of G-S. Confirming a conjecture of Brouwer, Gu [SIAM J. DiscreteMath. 35 (2021) 948--952] proved a tight lower bound on toughness of regulargraphs in terms of the second largest absolute eigenvalue. Fan, Lin and Lu[European J. Combin. 110 (2023) 103701] then studied the toughness of simplegraphs from the spectral radius perspective. While the toughness is animportant concept in graph theory, it is also very interesting to study |S| forwhich c(G-S)\geq l for a given integer l\geq 2. This leads to the concept ofthe l-connectivity, which is defined to be the minimum number of vertices of Gwhose removal produces a disconnected graph with at least l components or agraph with fewer than l vertices. Gu [European J. Combin. 92 (2021) 103255]discovered a lower bound on the l-connectivity of regular graphs via the secondlargest absolute eigenvalue. As a counterpart, we discover the connectionbetween the l-connectivity of simple graphs and the spectral radius. We alsostudy similar problems for digraphs and an edge version.

l-connectivity, l-edge-connectivity and spectral radius of graphs