We develop a new refinement of the Kato’s inequality and using this refinement we obtain several upper bounds for the numerical radius of a bounded linear operator as well as the product of operators, which improve the well known existing bounds. Further, we obtain a necessary and sufficient condition for the positivity of 2×2 certain block matrices and using this condition we deduce an upper bound for the numerical radius involving a contraction operator. Furthermore, we study the Schatten p-norm inequalities for the sum of two n×n complex matrices via singular values, and from the inequalities we obtain the p-numerical radius and the classical numerical radius bounds. We show that for every p>0, the p-numerical radius wp(·):Mn(C)→R satisfies wp(T)≤12|T|2(1-t)+|T∗|2(1-t)‖|T|2t+|T∗|2t‖p/2 for all t∈[0,1]. Considering p→∞, we get a nice refinement of the well known classical numerical radius bound w(T)≤12T∗T+TT∗. As an application of the Schatten p-norm inequalities we develop a bound for the energy of a graph. We show that E(G)≥2mmax1≤i≤n∑j,vi∼vjdj, where E(G) is the energy of a simple graph G with m edges and n vertices v1,v2,…,vn such that degree of vi is di for each i=1,2,…,n.