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.

A total labeling of simple and connected graph G(V, E) is said to be local antimagic total edge labeling if a bijection f : V(G) ∪ E(G) → {1, 2, 3, …, |V(G)| + |E(G)|}, w_t (e₁) ≠ w_t (e₂) for any two adjacent edges e₁ and e₂, where for e = uv ∈ G, w_t (e) = f (u) + f (uv) + f (v). The local antimagic total edge labeling induces a proper edge coloring of G if each edge e is assigned the color w_t(e). The minimum number of local antimagic total edge chromatic number of G denoted by γ_leat(G), is the distinct induced by edge labels over all local antimagic total labeling of G. In this paper we study the existence of local edge antimagic total chromatic number of amalgamation of some special graphs namely amal(P_n, v, m), amal(C_n, v, m),amal(F_n, v, m) and amal(S _n, v, m).