Let G(V,E) be a finite, undirected, simple graph without isolated vertices. A dominating set D of V(G) is called degree coprime G if for every v∈V-D, there exist vertex u∈D such that uv∈E(G) and (deg u,deg v)=1. The minimum cardinality the domination number denoted by γ_cp (G). non-coprime v)≠1. γ_ncp We obtain some graphs.

Let G_1 and G_2 be simple connected graphs with disjoint vertex sets V(G_1) and V(G_2), respectively. For given vertices a_1in V(G_1) and a_2in V(G_2), a splice of G_1 and G_2 by vertices a_1 and a_2 is defined by identifying the vertices a_1 and a_2 in the union of G_1 and G_2. In this paper, we present exact formulas for computing some vertex-degree-based graph invariants of splice of graphs.

Given a graph G $G$ , decomposition of is partition its edges. A ( d h ) $(d,h)$ -decomposable if edge set can be partitioned into $d$ -degenerate and with maximum degree at most $h$ . For ≤ 4 $d\le 4$ we are interested in the minimum integer ${h}_{d}$ such that every planar $(d,{h}_{d})$ -decomposable. It was known 3 ${h}_{3}\le 2 8 ${h}_{2}\le 8$ 1 = ∞ ${h}_{1}=\infty $ This paper proves ${h}...

the reciprocal degree distance (rdd)‎, ‎defined for a connected graph $g$ as vertex-degree-weighted sum of the reciprocal distances‎, ‎that is‎, ‎$rdd(g) =sumlimits_{u,vin v(g)}frac{d_g(u)‎ + ‎d_g(v)}{d_g(u,v)}.$ the reciprocal degree distance is a weight version of the harary index‎, ‎just as the degree distance is a weight version of the wiener index‎. ‎in this paper‎, ‎we present exact formu...