Let G = (V (G), E(G)) be a finite simple graph with p = |V (G)| vertices and q = |E(G)| edges,without isolated vertices or isolated edges. A vertexmagic total labeling is a bijection from V (G) ∪ E(G) to the consecutive integers 1, 2, . . . , p + q, with the property that, for every vertex u in V (G), one has f (u) +  uv∈E(G) f (uv) = k for some constant k. Such a labeling is called E-super ve...

Let G=(V,E) be a simple graph. A vertex labeling f:V(G)→{1,2,⋯,k} is defined to local inclusive (respectively, non-inclusive) d-distance irregular of graph G if for any two adjacent vertices x,y∈V(G) their weights are distinct, where the weight x∈V(G) sum all labels whose distance from x at most d but least 1). The minimum k which there exists called irregularity strength G. In this paper, we p...

the total version of geometric–arithmetic index of graphs is introduced based on the endvertexdegrees of edges of their total graphs. in this paper, beside of computing the total gaindex for some graphs, its some properties especially lower and upper bounds are obtained.

‎let $g$ be a graph with vertex set $v(g)$ and edge set $x(g)$ and consider the set $a={0,1}$‎. ‎a mapping $l:v(g)longrightarrow a$ is called binary vertex labeling of $g$ and $l(v)$ is called the label of the vertex $v$ under $l$‎. ‎in this paper we introduce a new kind of graph energy for the binary labeled graph‎, ‎the labeled graph energy $e_{l}(g)$‎. ‎it depends on the underlying graph $g$...

It is an elementary exercise to show that any non-trivial simple graph has two vertices with the same degree. This is not the case for digraphs and multigraphs. We consider generating irregular digraphs from arbitrary digraphs by adding multiple arcs. To this end, we define an irregular labeling of a digraph D to be an arc labeling of the digraph such that the ordered pairs of the sums of the i...

A total labeling of a graph with v vertices and e edges is defined as a one-to-one map taking the vertices and edges onto the integers 1, 2, · · · , v+e. Such a labeling is vertex magic if the sum of the label on a vertex and the labels on its incident edges is a constant independent of the choice of vertex, and edge magic if the sum of an edge label and the labels of the endpoints of the edge ...

Let G = (V,E) be a (p, q)-graph of order p and size q and f be a bijection from the set V ∪ E to the set of the first p + q natural numbers. The weight of a vertex is the sum of its label and the labels of all adjacent edges. We say f is an (a, d)-vertex-antimagic total labeling if the vertex-weights form an arithmetic progression with the initial term a and the common difference d. Such a labe...

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 ...

Let G = (V,E) be a finite (non-empty) graph, where V and E are the sets of vertices and edges of G. An edge magic total labeling is a bijection α from V ∪E to the integers 1,2, . . . , n+e, with the property that for every xy ∈E, α(x)+α(y)+α(xy)= k, for some constant k. Such a labeling is called an a-vertex consecutive edge magic total labeling if α(V )= {a + 1, . . . , a + n} and a b-edge cons...

A graph is called supermagic if there is a labeling of edges where the edges are labeled with consecutive distinct positive integers such that the sum of the labels of all edges incident with any vertex is constant. A graph G is called degree-magic if there is a labeling of the edges by integers 1, 2, ..., |E(G)| such that the sum of the labels of the edges incident with any vertex v is equal t...