Abstract :
Let G=(V,E) be a simple graph on vertex set V={v1,v2,…,vn}. Further let di be the degree of vi and Ni be the set of neighbors of vi. It is shown that is an upper bound for the largest eigenvalue of the Laplacian matrix of G, where Ni∩Nj denotes the number of common neighbors between vi and vj. 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 where
