Let G=(V,E) be the simple connected graph of order n and λ1(G) be the spectral radius of signless Laplacian of G. For each vi ∈ V(G),the set of its neighbors in G is denoted by Ni,and the number of neighbors of vi in G is denoted by di. Thte maximum degree of G is definedd by Δ(G). In this paper,it is showm that if G is connected graph and Δ(G)<n-1,then λ1(G)≥max{mi'+(1+(mi'-1)2/d2,i)di/mi':vi∈V(G) },where mi'=∑vivj∈E(G)(dj-|Ni∩Nj|)/di and d2,i is the number of vertices at distance two from vi in G. Also it is shown that λ1(G)≥max{(pij+(1-pij)2/pij max{1,di-1})|Ni∪Nj|:vivj∈E(G),di ≥ dj},Where pij=e(Ni,Nj-Ni)/di(|Ni∪Nj|-di). Then it is also shown that if G is a connected graph,then λ1(G)≤√2Δ(G)(Δ(G)-1)+1+1.
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