The G be a connected graph with vertex set V ( G ) and edge set E ( G ) . A subset S ⊆ V ( G ) is called a dominating set of G if for every vertex x in V ( G ) ∖ S , there exists at least one vertex u in S such that x is adjacent to u. An ordered set W ⊆ V ( G ) is called a resolving set of G, if every pair of vertices u and v in V ( G ) have distinct representation with respect to W. An ordered set S ⊆ V ( G ) is called a dominant resolving set of G, if S is a resolving set and also a dominating set of G. The minimum cardinality of dominant resolving set is called a dominant metric dimension of G, denoted by D d i m ( G ) . In this paper, we investigate the dominant metric dimension of some particular class of graphs, the characterisation of graph with certain dominant metric dimension, and the dominant metric dimension of joint and comb products of graphs.

Let $G=(V,E)$ be a graph with vertex set V and edge set E. A local antimagic total vertex coloring f of a graph G with vertex-set V and edge-set E is an injective map from $V\cup E$ to $\{1,2,\dots ,|V|+|E\left|\right\}$ such that if for each $uv\in E\left(G\right)$ then $w\left(u\right)\ne w\left(v\right)$, where $w\left(u\right)={\sum }_{uv\in E\left(G\right)}f\left(uv\right)+f\left(u\right)$. If the range set f satisfies $f\left(V\right)=\{1,2,\dots ,|V\left|\right\}$, then the labeling is said to be local super antimagic total labeling. This labeling generates a proper vertex coloring of the graph G with the color $w\left(v\right)$ assigning the vertex v. The local super antimagic total chromatic number of graph G, ${\chi }_{lsat}\left(G\right)$ is defined as the least number of colors that are used for all colorings generated by the local super antimagic total labeling of G. In this paper we investigate the existence of the local super antimagic total chromatic number for some particular classes of graphs such as a tree, path, cycle, helm, wheel, gear, sun, and regular graphs as well as an amalgamation of stars and an amalgamation of wheels.

Let G = (V, E) be a simple, nontrivial, finite, connected and undirected graph. For an integer 1 ≤ k ≤ diam(G), a distance k-dominating set of a connected graph G is a set S of vertices of G such that every vertex of V (G)\S is at distance at most k from some vertex of S. The k-domination number, denoted by γ _k (G), of G is the minimum cardinality of a k-dominating set of G. In this paper, we improve the lower bound on the distance domination number of G regarding to the diameter and minimum degree as well as the upper bound regarding to the order and minimum k distance neighbourhood. In addition, we determine the bound of distance domination number of edge comb product graph.

The local antimagic labeling on a graph G with |V| vertices and |E| edges is defined to be an assignment f : E --> {1, 2,..., |E|} so that the weights of any two adjacent vertices u and v are distinct, that is, w(u)̸ ̸= w(v) where w(u) = Σe∈E(u) f(e) and E(u) is the set of edges incident to u. Therefore, any local antimagic labeling induces a proper vertex coloring of G where the vertex u is assigned the color w(u). The local antimagic chromatic number, denoted by χla(G), is the minimum number of colors taken over all colorings induced by local antimagic labelings of G. In this paper, we present the local antimagic chromatic number of unicyclic graphs that is the graphs containing exactly one cycle such as kite and cycle with two neighbour pendants.

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.

Let G = (V(G), E(G)) be a nontrivial connected graph with an edge coloring c : E(G) → {1, 2, ..., l}, l N, with the condition that the adjacent edges may be colored by the same colors. A path P in G is called rainbow path if no two edges of P are colored the same. The smallest number of colors that are needed to make G rainbow edge-connected is called the rainbow edge-connection of G, denoted by rc(G). A vertex-colored graph is rainbow vertex-connected if any two vertices are connected by a path whose internal vertices have distinct colors. The smallest number of colors that are needed to make G rainbow vertex-connected is called the rainbow vertex-connection of G, denoted by rvc(G). A total-colored path is total-rainbow if edges and internal vertices have distinct colours. The minimum number of colour required to color the edges and vertices of G is called the total rainbow connection number of G, denoted by trc(G). In this paper, we determine the total rainbow connection number of some wheel related graphs such as gear graph, antiweb-gear graph, infinite class of convex polytopes, sunflower graph, and closed-sunflower graph.