Inequalities for real number sequences with applications in spectral graph theory

Let a = (a1,a2, …, an) be a nonincreasing sequence of positive real numbers. Denote by S = {1, 2, …, n} the index set and by Jk = {I = {r1, r2, …, rk}, 1 ⩽ r1 < r2 < … < rk ⩽ n} the set of all subsets of S of cardinality k, 1 ⩽ k ⩽ n − 1. In addition, denote by aI=ar1+ar2+…+ark,1⩽k⩽n−1,1⩽r1<r2<…<rk⩽n, the sum of k arbitrary elements of sequence a, where aI1=a1+a2+…+ak and aIn=an−k+1+an−k+2+…+an. We consider bounds of the quantities RSk(a)=aI1/aIn,LSk(a)=aI1−aIn and Sk,α(a)=∑I∈JkaIα in terms of A=∑i=1nai and B=∑i=1nai2. Then we use the obtained results to generalize some results regarding Laplacian and normalized Laplacian eigenvalues of graphs.