11 Annals of Pure and Applied Mathematics Vol. 8, No. 1, 2014, 11-14 ISSN: 2279-087X (P), 2279-0888(online) Published on 14 October 2014 www.researchmathsci.org Spectral Conditions for a Graph to be k-Connected Rao Li Department of Mathematical Sciences University of South Carolina Aiken Aiken, SC 29801, USA, Email: raol@usca.edu Received 20 September 2014; accepted 28 September 2014 Abstract. Using spectral radius and signless Laplacian spectral radius, we in this note present sufficient conditions for a graph to be k -connected. Keywords: k-connected graph, spectral radius, signless Laplacian spectral radius AMS Mathematics Subject Classification (2010): 05C40, 05C50 1. Introduction We consider only finite undirected graphs without loops or multiple edges. Notation and terminology not defined here follow those in [2]. For a graph =( , ) G VE , we use n and e to denote its order | | V and size | | E , respectively. We use         to denote the degree sequence of a graph. The eigenvalues of a graph G are defined as the eigenvalues of its adjacency matrix ( ) AG . The largest eigenvalue, denoted ( ) G ρ , of a graph G is called the spectral radius of G . The signless Laplacian eigenvalues of a graph G are defined as the eigenvalues of the matrix ( ) := ( ) ( ) QG DG AG + , where ( ) DG is the diagonal matrix 1 2 ( , ,..., ) n diag d d d and ( ) AG is the adjacency matrix of G . The largest signless Laplacian eigenvalue, denoted ( ) qG , of a graph G is called the signless Laplacian spectral radius of G . 2. Main results In [4], Li obtained sufficient conditions which are based on the spectral radius for some Hamiltonian properties of graphs. In [5], Li obtained sufficient conditions which are based on the signless Laplacian spectral radius for some Hamiltonian properties of graphs. Using similar ideas as the ones in [4] and [5], we will present sufficient conditions which are based on the spectral radius or the signless Laplacian spectral radius for a graph to be k - connected. The results are as follows. Theorem 1. Let G be a connected graph of order 2 n ≥ and let 1 1 k n ≤ ≤ - . If 2 5 ( 5) 8( 3 ( 1)( 1)) > , 4 n k n k n k n k ρ + - + + - + + - + - - ∆- then G is k - connected. Theorem 2. Let G be a connected graph of order 2 n ≥ and let 1      1. If  