An improved upper bound for Laplacian graph eigenvalues

Let G=(V,E) be a simple graph on vertex set V={v ₁,v ₂,⋯,v _n}. Further let d _i be the degree of v _i and N _i be the set of neighbors of v _i. It is shown that max {d _i+d _j-|N _i∩N _j|:1i<jn, vivj ∈ E} is an upper bound for the largest eigenvalue of the Laplacian matrix of G, where |N _i∩N _j| denotes the number of common neighbors between v _i and v _j. For any G, this bound does not exceed the order of G. Further using the concept of common neighbors another upper bound for the largest eigenvalue of the Laplacian matrix of a graph has been obtained as max {√2(d ₁ ²+d _im' _i):1 ≤ i ≤ n}, where m' _i = ∑ _j{d _j-|N _i∩N _j|:vivj ∈ E}/d _i.