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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 fe from the edge set to the first ke natural number, and a function fv from the vertex set to the non negative even number up to 2kv, where k = max{ke , 2kv }. 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.

Let G(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 and u’v’ in path x — y,w(uv) = w(u’v’) w(uv), where w(uv) = f (u) + f (v) and x,y ∈ V(G). A graph G is a rainbow antimagic connection if G has a rainbow antimagic labeling. Thus any rainbow antimagic labeling induces a rainbow coloring of G where the edge uv is assigned with the color w(uv). The rainbow antimagic connection number of G, denoted by rac(G), is the smallest number of colors taken over all rainbow colorings induced by rainbow antimagic labeling of G. In this paper, we show the exact value of the rainbow antimagic connection number of jahangir graph J2,m, lemon graph Lem, firecracker graph (Fm,3), complete bipartite graph (K2,m), and double star graph (Sm,m).

A rainbow antimagic coloring is one of new topics in graph theory. This topic is an expansion of rainbow coloring that is combined with antimagic labeling. The graphs are labeled with an antimagic labeling, and then the sum of vertex label have to obtain a rainbow coloring. The aim of the rainbow antimagic coloring research is to find the minimum number of color, called rainbow antimagic connection number, denoted by rac(G). In this research, we studied some simple graphs to be colored with rainbow antimagic coloring. The graphs we used are lollipop, stacked book, Dutch windmill, flowerpot and dragonfly. In this research, we aimed to develop new theorem. Based on the results, we got some theorems about rac(G) for the rainbow antimagic coloring. In addition, on some graphs that we use, we get rac(G) equal to rc(G), rainbow connection number. Instead, on the others, the value of rac(G) cannot reach the rc(G).

Let G(V, E) be a connected and simple graphs with vertex set V and edge set E. Define a coloring c : E(G) → {1, 2, 3, …, k}, k ∈ N as the edges of G, where adjacent edges may be colored the same. If there are no two edges of path P are colored the same then a path P is a rainbow path. The graph G is rainbow connected if every two vertices in G has a rainbow path. A graph G is called antimagic if the vertex sum (i.e., sum of the labels assigned to edges incident to a vertex) has a different color. Since the vertex sum induce a coloring of their edges and there always exists a rainbow path between every pair of two vertices, we have a rainbow antimagic coloring. The rainbow antimagic connection number, denoted by rcA (G) is the smallest number of colors that are needed in order to make G rainbow connected under the assignment of vertex sum for every edge. We have found the exact value of the rainbow antimagic connection number of ladder graph, triangular ladder, and diamond.

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All graphs in this paper are nontrivial and connected simple graphs. For a set W = {s1,s2,...,sk} of verticesof G, the multiset representation of a vertex v of G with respect to W is r(v|W) = {d(v,s1),d(v,s2),...,d(v,sk)} whered(v,si) is the distance between of v and si. If the representation r(v|W)̸= r(u|W) for every pair of vertices u,v of a graph G, the W is called the resolving set of G, and the cardinality of a minimum resolving set is called the multiset dimension, denoted by md(G). A set W is a local resolving set of G if r(v|W) ̸= r(u|W) for every pair of adjacent vertices u,v of a graph G. The cardinality of a minimum local resolving set W is called local multiset dimension, denoted by µl(G). In our paper, we discuss the relationship between the multiset dimension and local multiset dimension of graphs and establish bounds of local multiset dimension for some families of graph.

The study of metric dimension of graph G has widely given some results and contribution of graph research of interest, including the domination set theory. The dominating set theory has been quickly growing and there are a lot of natural extension of this study, such as vertex domination, edge domination, total domination, power domination as well as the strong domination. In this study, we initiate to combine the two above concepts, namely metric dimension and strong domination set. Thus we have a resolving strong domination set. We have obtained the resolving strong domination number, denoted by γrst(G), of some graphs.