The notion of sum graphs was rst introduced by Harary [7]. From a practical point of view, sum **graph** **labeling** can be used as a compressed representation of a **graph**, a data structure for representing the **graph**. Data compression is important not only for saving memory space but also for speeding up some **graph** algorithms when adapted to work with the compressed representation of the input **graph** (for example, see [5,10]). There have been several papers determining or bounding the sum number of partic- ular classes of graphs G = (V; E) (n = |V |, m = |E|):

Let G = (V, E) be a finite simple and undirected graphs. That is to say, they contain neither loops nor multiple edges. The **graph** have vertex- set V (G) and edge-set E(G), and we denote |V (G)| and |E(G)| by p and q respectively. **Graph** **Labeling** is giving a label at vertices and edges of **graph**, so that each vertices and each edges having different label. Generally, label uses positive integer.

Gallian [2] defined a **graph** **labeling** as an assignment of integers to the vertices or edges, or both, subject to certain condition. Magic labelings was first introduced in 1963 by Sedl´aˇck [9]. The concept of H-magic graphs was introduced in [3]. An edge-covering of a **graph** G is a family of different subgraphs H 1 , H 2 , ..., H k such that each edge of E belongs to at least

When I first heard of the idea of a dynamic survey (that is, an electronic paper that is occasionally updated without undergoing further review by referees and available free on the web) I thought **graph** **labeling** was a perfect candidate– the field was growing rapidly with papers appearing in journals from around the world. In my initial dynamic article I updated my 1994 survey and expanded the range of labelings included. Although I submitted it to the Electronic Journal of Combinatorics (EJC) in September 1996 it was not accepted until November 1997 [14]. In fact, about a year after submitting it I received a letter from the editor of EJC saying they were rejecting it because they could not find anyone to agree to referee it. I wrote the editor back suggesting several people who I thought might be willing to referee it. One of them was Alex Rosa, who had been helpful to me in some earlier occasions about where to submit **labeling** papers. Of course, I was not told who finally agreed to serve as referee but it was accepted a few months later. One obvious problem with a dynamic survey is that the journal does not retain a permanent copy of earlier versions. It did not occur to me to save copies of previous editions so I do not know how many pages the first edition had nor the number of references it included. The same is true for several later editions. Following is the only information I have about the various editions.

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Gallian [2] defined a **graph** **labeling** as an assignment of integers to the vertices or edges, or both, subject to certain condition. **Graph** **labeling** was first introduced in the late 1960s. Magic **labeling** is a type of **graph** **labeling** that the most often to be studied.

One type of **graph** **labeling** problem involves **labeling** the vertices of a **graph** G and then computing a value g(v) for each v ∈ V (G), where g(v) is determined by the labels on some set S(v) ⊂ V (G). Properties of the **graph** can be defined based on the permissible sets of values that are produced by the set of labelings {g(v) : v ∈ V (G)}. For example, for a **graph** G = (V, E) of order n, one can define a bijection f : V (G) → {1, 2, . . . , n} and then for each vertex, sum the labels in its open (or closed) neighborhood. One case that has been studied is the case where the set of resulting open neighborhood sums are all equal. Vilfred [9] called such a **labeling** a Σ **labeling** and any **graph** for which such a **labeling** exists a Σ **graph**. Miller et al. [2] referred to such a **labeling** as a 1-vertex magic **labeling**. More recently Sugeng et al. [8] have referred to such a **labeling** as a distance magic **labeling**. When the closed neighborhood sums are all equal, Beena [1] has referred to the **labeling** as a Σ ′

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[3] Balbuena, C., Barker, E., Das, K.C., Lin, Y., Miller, M., Ryan, J., Slamin, Sugeng, K., Tk´ aˇ c, M., ”On the degrees of a strongly vertex-magic **graph**”, Discrete Math., 306 (2006), 539–551. [4] Gallian, J.A., ”A dynamic survey of **graph** **labeling**”, Electron. J. Combin., 6 (2010) DS 6. [5] G´ omez, J., ”Solution of the conjecture: If n ≡ 0 (mod 4), n > 4, then K n has a super

Gallian Department of Mathematics and Statistics University of Minnesota Duluth, Duluth, MN 55812, USA Abstract.In this article I trace my involvement with graph labeling for the past[r]

graf adalah suatu cabang dari matematika yang pada akhir-akhir ini ber- kembang pesat. Salah satu bagian dari graf yang banyak diminati adalah **graph** **labeling**. **graph** **labeling** adalah pemberian nilai (integer positif) pada vertex, edge, atau vertex dan edge. Ada beberapa macam **labeling**, yaitu

The **graph** **labeling** can also be defined as different function (Gallian, 2011) . For a **graph** G of order n and size m, a γ -**labeling** of G is an one-to-one function f: V(G) → {0, 1, 2, …m} that induces a **labeling** f’: E(G) → {1, 2, 3, …, m} of the edges of G defined by f’(e) = | f(u) – f(v) | for each edge e = uv of G. Chartrand et al. (2005) showed that every connected **graph** has a γ -**labeling**. Each γ -**labeling** f of **graph** G of order n and size m is assigned a value denoted by val(f) and defined by val(f)= Σ e ε E f’(e).

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In this paper we consider simple, finite, connected and undirected **graph**. A **graph** 𝐺(𝑝, 𝑞) with 𝑝 = |𝑉(𝐺)| vertices and 𝑞 = |𝐸(𝐺)| edges. A **graph** **labeling** which has often been motivated by practical problems is one of fascinating areas of research. Labeled graphs serves as useful mathematical models for many applications in coding theory, communication networks, and mobile telecommunication system. We refer to Gallian [1] for a dynamic survey of various **graph** **labeling** problems along with extensive bibliography. Most **graph** **labeling** methods trace their origin to one introduced by Rosa in 1967, or one given by Graham and Sloane in 1980.

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Pelabelan sisi ajaib pada **graph** G(V, E) dengan banyak titik p dan banyak sisi q adalah fungsi bijektif f: V ∪ E 1,2, … , p + q dan untuk setiap u, v ϵ E(G) berlaku f u + f u, v + f v = s , dengan v merupakan titik yang terhubung langsung dengan titik u, dan s merupakan konstanta ajaib pada **graph** G. Selanjutnya **graph** G(p, q) disebut **graph** ajaib sisi jika terdapat pelabelan sisi ajaib pada **graph** tersebut. Suatu **graph** sisi ajaib dikatakan super sisi ajaib jika terdapat fungsi bijektif f: (V G ) 1,2, … , p . Tugas akhir ini akan membahas mengenai beberapa kelas **graph** planar super sisi ajaib, yang diantaranya ialah **graph** P 2n (+)N m , **graph** (P 2 ∪ kK 1 ) + N 2 , **graph** payung, **graph** kelabang, **graph** cumi-cumi, **graph**

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Jenis-Jenis Graph Berdasarkan jumlah simpul pada suatu graph, maka secara umum graph dapat digolongkan menjadi dua jenis: 1.. Graph berhingga limited graph 2.[r]

The concept of H-magic graphs was introduced by Guti´errez and Llad´o [4] in 2005. Suppose G = (V, E) admits an H-covering. We say that a bijective function f : V (G) ∪ E(G) → {1, 2, . . . , |V (G)| + |E(G)|} is an H-magic **labeling** of G if there exists a positive integer m(f ), called magic sum, such that for any subgraph H ′

Selanjutnya Levin dkk 2000 menyebutkan bahwa hal yang terpenting untuk menjelaskan kejahatan adalah dengan labeling theory, dinamakan begitu karena teori labeling berpendapat bahwa kel[r]

The second main step in the CVD algorithm is swapping the two colos along the chain. Kempe chain arguments is a technique that can be used by swapping the colors in one H(i, j). Swapping can be used to free up a color somewhere. It does not create a new clnflict but it moves the confliciting vertex along the path. The algorithm will stop the color swapping when 1) another conflicting vertex is reached; 2) where the conflict may cancel out, or 3) the number of swaps reaches a given limit. We set the given limit at most the number of vertices in the **graph**. The algorithm keeps running until one of these condition is reached:1) there is no conflicting vertex anymore or 2) all conflics are solved (getting a 3-edge colored), or 3) it reaches fixed limit.

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The reseacrh aims to reveal the understanding of adolescents about a **labeling** that received clearly. This reasearch based on many label received by them from society and focused on how they interpret the label received from society whether positive or negative through what they feel, think, wish for, did for and about believe in a label received. The research used 4 subjects and the method used is qualitative phenomenology with the technique of collecting data that was a semi-structured interview. The process of validity based on certainly whether the research results have a proper results from the prospective of researchers, participation or the reader. The research results show that adolescents in interpreting the label as a negative thing will tend to ignore and the labels not affect the behavior, because they was sure that not important to change self identity. The adolescent that interpret label as a positive thing tend to believe with label because they think that label is a valuation from society a nd assume a label as a daily behavior guideline.

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[5] D. Suryadi H.S., “Pengantar Teori dan Algoritma **Graph**”, Gunadarma, Jakarta [6] Mary E.S. Loomis, “Data Structure and File Processing”, Prentice-Hall, [7] Suryadi MT, “Pengantar Analisis Algoritma”, Gunadarma, Jakarta, 1992 Keterangan :