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Let $G = (V, E)$ be a connected graph and $m$ be a positive integer, the conditional edge connectivity $\lambda^m_\delta$ is the minimum cardinality of a set of edges, if it exists, whose deletion disconnects $G$ and leaves each remaining component with minimum degree $\delta$ no less than $m.$ This study shows that $\lambda^1_\delta (Q_{n,k}) = 2n,$ $λ^2_\delta(Q_{n,k}) = 4n − 4$$(2 ≤ k ≤ n − 1, n ≥ 3)$ for $n$-dimensional enhanced hypercube $Q_{n,k}.$ Meanwhile, another easy proof about $\lambda^2_\delta (Q_n) = 4n − 8,$ for $n ≥ 3$ is proposed. The results of enhanced hypercube include the cases of folded hypercube.
}, issn = {}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/aam/20580.html} }Let $G = (V, E)$ be a connected graph and $m$ be a positive integer, the conditional edge connectivity $\lambda^m_\delta$ is the minimum cardinality of a set of edges, if it exists, whose deletion disconnects $G$ and leaves each remaining component with minimum degree $\delta$ no less than $m.$ This study shows that $\lambda^1_\delta (Q_{n,k}) = 2n,$ $λ^2_\delta(Q_{n,k}) = 4n − 4$$(2 ≤ k ≤ n − 1, n ≥ 3)$ for $n$-dimensional enhanced hypercube $Q_{n,k}.$ Meanwhile, another easy proof about $\lambda^2_\delta (Q_n) = 4n − 8,$ for $n ≥ 3$ is proposed. The results of enhanced hypercube include the cases of folded hypercube.