Vertex-connectivity and eigenvalues of graphs with fixed girth

Abstract Let κ ( G ), g ( G ), δ ( G ) and Δ( G ) denote the vertex-connectivity, the girth, the minimum degree and the maximum degree of a simple graph G , and let λ i ( G ), μ i ( G ) and q i ( G ) denote the i th largest adjacency eigenvalue, Lapalcian eigenvalue and signless Laplacian eigenvalue of G . We investigate functions f ( δ , Δ, g, k ) with Δ ≥  δ  ≥  k  ≥ 2 and g  ≥ 3 such that any graph G satisfying λ 2 ( G )  f ( δ ( G ), Δ( G ), g ( G ), k ) has connectivity κ ( G ) ≥  k . Analogues results involving the Laplacian eigenvalues and the signless Laplacian eigenvalues to describe connectivity of a graph are also presented. As corollaries, we show that for an integer k  ≥ 2 and a simple graph G with n = | V ( G ) | , maximum degree Δ and minimum degree δ  ≥  k , the connectivity κ ( G ) ≥  k if one of the following holds. (i) λ 2 ( G ) δ − ( k − 1 ) Δ n 2 ( δ − k + 2 ) ( n − δ + k − 2 ) , or (ii) μ n − 1 ( G ) > ( k − 1 ) Δ n 2 ( δ − k + 2 ) ( n − δ + k − 2 ) , or (iii) q 2 ( G ) 2 δ − ( k − 1 ) Δ n 2 ( δ − k + 2 ) ( n − δ + k − 2 ) .