Generalizations of Szőkefalvi Nagy and Chebyshev inequalities with applications in spectral graph theory

Two weighted inequalities for real non-negative sequences are proven. The first one represents a generalization of the Szőkefalvi Nagy inequality for the variance, and the second a generalization of the discrete Chebyshev inequality for two real sequences. Then, the obtained inequalities are used to determine lower bounds for some degree-based topological indices of graphs.