LetG(V, E)be a connected, undirected and simple graph with vertex set V(G) and edge set E(G). A labeling of a graph G is a bijection f from V(G) to the set {1, 2,…, | V(G)|}. The bijection f is called rainbow antimagic vertex labeling if for any two edge uv andu’v’in pathx — y,w(uv)=w(u’v’)w(uv),wherew(uv)= f (u) + f (v) andx,y ∈V(G). A graph G is a rainbow antimagic connection ifGhas a rainbow antimagic labeling. Thus any rainbow antimagic labeling induces a rainbow coloring ofGwhere the edgeuvis assigned with the color w(uv). The rainbow antimagic connection number of G, denoted byrac(G),is the smallest number of colors taken over all rainbow colorings induced by rainbow antimagic labeling ofG. In this paper, we show the exact value of the rainbow antimagic connection number of jahangir graph J₂,_m, lemon graphLe_m,firecracker graph(F_m,3), complete bipartite graph (K₂,_m), and double star graph (S_m,_m).

Let G be a connected, simple, and undirected graph with a vertex set V(G) and an edge set E(G). Total k-labeling is a function f_e from the edge set to the first k_e natural number, and a function f_v from the vertex set to the non negative even number up to 2k_v, where k = max{k_e , 2k_v }. An edge irregular reflexive k labeling of the graph G is the total k-labeling, if for every two different edges x ₁ x ₂ and x 1 ′ x 2 ′ of G , w t ( x 1 x 2 ) ≠ w t ( x 1 ′ x 2 ′ ) , where w t ( x 1 x 2 ) = f v ( x 1 ) + f e ( x 1 x 2 ) + f v ( x 2 ) . The minimum k for graph G which has an edge irregular reflexive k-labelling is called the reflexive edge strength of the graph G, denoted by res(G). In this paper, we determined the exact value of the reflexive edge strength of family trees, namely generalized sub-divided star graph, broom graphs, and double star graph.

A function f with domain and range are respectively the edge set of graph G and natural number up to ke, and a function f with domain and range are respectively the vertex set of graph G and the even natural number up to 2kv are called a total k-labeling where k=max{ke,2kv}. The total k-labeling of graph G by the condition that every two different edges have different weight is called an edge irregular reflexive k-labeling, where for any edge x1x2, the weight is wt(x1x2)=fv(x1)+fe(x1x2)+fv(x2). The reflexive edge strength of the graph G, denoted by res(G) is the minimum k for graph G which has an edge irregular reflexive k-labelling. In this study, we obtained the res(G) of graphs which their vertex degrees show an almost regularity properties.

The dynamic coloring of a graph G is proper coloring such that every vertex of G with degree has at least two neighbors that are colored differently. A generalization of the dynamic coloring was also introduced by Montgomery in [12], the generalized concept is called r-dynamic k-coloring. An r-dynamic coloring of a graph G is a proper coloring c of the vertices such that |c(N(v)| ≥ min{r, d(v)}, for each v ∈V(G). The r-dynamic chromatic number of a graph G, denoted χ_r(G) is the smallest k such that c is an r-dynamic k coloring of G. We will find the lower bound of the r-dynamic chromatic number of graphs corona wheel graph and some new results the exact value of r-dynamic chromatic number of corona graphs. In this paper, we study the lower bound of X_r(H⊙W_m), X_r(W_n⊙H) and we also prove the exact value of r-dynamic chromatic number of some graphs.