Let G = (V,E) be a graph of order n and let B(S) be the set of vertices in V\S that have a neighbor in the vertex set S. The differential of a vertex set S is defined as ?(S) = |B(S)| - |S| and the maximum value of ?(S) for any subset S of V is the differential of G. A Roman dominating function of G is a function f : V ? {0,1,2} such that every vertex u with f(u) = 0 is adjacent to a vertex v with f(v) = 2. The weight of a Roman dominating function is the value f(V) = ?u?V f(u). The minimum weight of a Roman dominating function of a graph G is the Roman domination number of G, written ?R(G). We prove that ?R(G) = n - ?(G) and present several combinatorial, algorithmic and complexity-theoretic consequences thereof.

At present, inequalities have reached an outstanding theoretical and applied development and they are the methodological base of many mathematical processes. In particular, Hermite– Hadamard inequality has received considerable attention. In this paper, we prove some new results related to Hermite–Hadamard inequality via symmetric non-conformable integral operators.