A Roman dominating function of a graph G is a labeling f : V (G) → {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2. The Roman domination number γR(G) of G is the minimum of ∑ v∈V (G) f(v) over such functions. Let G be a connected n-vertex graph. We prove that γR(G) ≤ 4n/5, and we characterize the graphs achieving equality. We obtain sharp upper and lower bounds for γR(...

Pell graphs are defined on certain ternary strings as special subgraphs of Fibonacci cubes odd index. In this work the domination number, total 2-packing connected paired and signed number studied. Using integer linear programming, exact values some estimates for these numbers small obtained. Furthermore, theoretical bounds obtained graphs.