A few properties of the eigenvalues of normalized graph Laplacian

Here we have investigated a few properties of the eigenvalues of normalized (geometric) graph Laplacian in different graphs. Preservation of eigenvalue 1 from a particular subgraph to the entire graph, the spectrum of the graph constructed with triangles share a common vertex have been addressed. Further using the number and degrees of common neighbors between vertices some new upper bounds for the largest eigenvalue have been introduced.