In this paper, we explore the 2-extra connectivity and 2-extra-edge-connectivity of the folded hypercube FQ n . We show that j 2 (FQ n ) = 3n À 2 for n P 8; and k 2 (FQ n ) = 3n À 1 for n P 5. That is, for n P 8 (resp. n P 5), at least 3n À 2 vertices (resp. 3n À 1 edges) of FQ n are removed to get a disconnected graph that contains no isolated vertices (resp. edges). When the folded hypercube is used to model the topological structure of a large-scale parallel processing system, these results can provide more accurate measurements for reliability and fault tolerance of the system.

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Nikiforov conjectured that for a given integer k ≥ 2, any graph G of sufficiently large order n with spectral radius µ(G) ≥ µ(S n,k ) (or µ(G) ≥ µ(S + n,k )) contains C 2k+1 or C 2k+2 (or C 2k+2 ), unless G = S n,k (or G = S + n,k ), where C ℓ is a cycle of length ℓ and S n,k = K k ∨ K n−k , the join graph of a complete graph of order k and an empty graph on n − k vertices, and S + n,k is the graph obtained from S n,k by adding an edge in the independent set of S n,k . In this paper, a weaker version of Nikiforov's conjecture is considered, we prove that for a given integer k ≥ 2, any graph G of sufficiently large order n with spectral radius µ(G) ≥ µ(S n,k ) (or µ(G) ≥ µ(S + n,k )) contains a cycle C ℓ with ℓ ≥ 2k + 1 (or C ℓ with ℓ ≥ 2k + 2), unless G = S n,k (or G = S + n,k ). These results also imply a result of Nikiforov given in [Theorem 2, The spectral radius of graphs without paths and cycles of specified length, LAA, 2010].

An extremal graph for a graph H on n vertices is a graph on n vertices with maximum number of edges that does not contain H as a subgraph. Let T n,r be the Turán graph, which is the complete r-partite graph on n vertices with part sizes that differ by at most one. The well-known Turán Theorem states that T n,r is the only extremal graph for complete graph K r+1 . Erdös et al. (1995) determined the extremal graphs for intersecting triangles and Chen et al. (2003) determined the maximum number of edges of the extremal graphs for intersecting cliques. In this paper, we determine the extremal graphs for intersecting odd cycles.

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