Computation on spectral radius of a graph

The molecular stability and related chemical properties are closely linked to the spectrum of the graph and corresponding eigenvalues. In quantum chemistry, spectral radius of graphs is the maximum energy level of molecules. Therefore, good upper bounds for the spectral radius is beneficial to estimate the energy of molecules. In this paper, we give two sharp upper bounds on the adjacency spectral radius of a graph in terms of degrees and the average 2-degrees of vertices. Moreover, we determine extremal graphs which achieve these upper bounds. Finally, some examples illustrate that the results are best in all known upper bounds in some sense.