Nowadays, the problem of finding families graphs for which one may ensure existence a vertex-labeling and/or an edge-labeling based on certain class integers, constitutes challenge researchers in both number and graph theory. In this paper, we focus those vertex-labelings whose induced multiplicative assigns hyper totient numbers to edges graph. way, introduce characterize notions re...

A vertex magic total (VMT) labeling of a graph G = (V,E) is a bijection from the set of vertices and edges to the set of integers defined by λ : V ∪E → {1, 2, . . . , |V | + |E|} so that for every x ∈ V , w(x) = λ(x)+ ∑ xy∈E λ(xy) = k, for some integer k. A VMT labeling is said to be a super VMT labeling if the vertices are labeled with the smallest possible integers, 1, 2, . . . , |V |. In thi...

Let G = (V,E) be a graph of order n and size e. An (a, d)-vertexantimagic total labeling is a bijection α from V (G) ∪ E(G) onto the set of consecutive integers {1, 2, . . . , n+ e}, such that the vertex-weights form an arithmetic progression with the initial term a and the common difference d. The vertex-weight of a vertex x is the sum of values α(xy) assigned to all edges xy incident to the v...

Graph theory contains several topics, one of which will be discussed in this study is graph labeling. In the topic labeling, used a limited, simple, and undirected graph. study, type labeling total The multistar combination star graphs whose center vertex are not connected to each other. This research uses literature method divided into two parts, that basic calculation determine boundary first...

An antimagic labeling of a graph with m edges and n vertices is a bijection from the set of edges to the integers 1, . . . , m such that all n vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with the same vertex. A graph is called antimagic if it has an antimagic labeling. In this paper we discuss antimagic properties of graphs which contain vert...

A finite directed labeled graph G with vertices {vi)~=l and labels {~)~i~ is solvable with delay d, if for every sequence of labels c h -.ok one can determine a sequence of vertices v% "" v~ such that there is a path connecting the k vertices vt ° ... v~ k labeled g~ ..~ , and for every j v~j is determined by the subsequence a~-+t ... oJ+a only. The paper presents a family of counterexamples pr...

A total labelling of a graph with v vertices and e edges is a one-to-one map taking the vertices and edges onto the set {1, 2, 3, . . . , v + e}. A labelling can be used to define a weight for each vertex and edge. For a vertex the weight is the sum of the label of the vertex and the labels of the incident edges. For an edge {x, y} the weight is the sum of the label of the edge and the labels o...