An edge e of a k-connected graph G is said to be a removable edge if G O e is still k-connected, where G e denotes the graph obtained from G by deleting e to get G - e, and for any end vertex of e with degree k - 1 in G- e, say x, delete x, and then add edges between any pair of non-adjacent vertices in NG-e （x）. The existence of removable edges of k-connected graphs and some properties of 3-connected and 4-connected graphs have been investigated [1, 11, 14, 15]. In the present paper, we investigate some properties of 5-connected graphs and study the distribution of removable edges on a cycle and a spanning tree in a 5- connected graph. Based on the properties, we proved that for a 5-connected graph G of order at least 10, if the edge-vertex-atom of G contains at least three vertices, then G has at least （3│G│ ＋ 2）/2 removable edges.
For a non-zero real number α, let sα（G） denote the sum of the αth power of thenon-zero Laplacian eigenvalues of a graph G. In this paper, we establish a connection betweensα（G） and the first Zagreb index in which the H¨older’s inequality plays a key role. By usingthis result, we present a lot of bounds of sα（G） for a connected （molecular） graph G in terms ofits number of vertices （atoms） and edges （bonds）. We also present other two bounds for sα（G）in terms of connectivity and chromatic number respectively, which generalize those results ofZhou and Trinajsti′c for the Kirchho？ index [B Zhou, N Trinajsti′c. A note on Kirchho？ index,Chem. Phys. Lett., 2008, 455： 120-123].
