(1) All tree **graphs** have a prime vertex labeling (Entringer-Tout Conjecture); (2) All unicyclic **graphs** have a prime vertex labeling (Seoud and Youssef [7]). In 2011, Haxell and Pikhurko [4] proved that all large trees are prime. In 1991, Deretsky et al. [2] introduced the notion of dual of prime labeling which is known as vertex prime labeling. A graph with q edges has vertex prime labeling if its edges can be labeled with distinct integers {1, 2, . . . , q} such that for each vertex of degree at least two the greatest common divisor of the labels on its incident edges is 1. For convenience, we will use [a, b] to denote the set of integers between a and b inclusively.

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Definition 2.1. [1] Let G be a simple graph of order n. Then R G ( ) be the graph obtained from G by adding a new vertex corresponding to each **edge** of G and by joining each new vertex to the end points, of the **edge** corresponding to it. It is called the **semi** total point graph.

any g ∈ G, s ∈ S. Thus the **edge** set consists of pairs of the form (g, gs), with s ∈ S providing the color. The set S is usually assumed to be finite, symmetric S = S −1 and the identity element of the group is excluded S. In this case, the uncolored Cayley graph is a simple undirected graph. We can consider S as a subset of non-identity elements G instead of being a generating set. A Cayley graph is connected if and only if G = hSi. In general the Cayley graph over the group hS i is a component of the main Cayley graph over the group G. There are many researches about the Cayley graph have been done by some authors for instance see [3, 9].

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Thus **semi**-transitive orientations generalize transitive orientations. A way to check if a given oriented graph G is **semi**-transitively ori- ented is as follows. First check that G is acyclic; if not, the orientation is not **semi**-transitive. Next, for a directed **edge** from a vertex x to a vertex y, consider each directed path P having at least three edges without repeated vertices from x to y, and check that the subgraph of G induced by P is transitive. If such non-transitive subgraph is found, the orientation is not **semi**-transitive. This procedure needs to be applied to each **edge** in G, and if no non-transitivity is discovered, G’s orientation is **semi**-transitive.

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Proof. Because is an **edge**-transitive graph, omitting each one of edges with its vertices makes isomorphism **graphs** .Using lemma 1, the **graphs** \ , that ∈ ( ) have the same number of perfect matchings. Now we’re counting the number of ( , ) where is a perfect matching of and e belongs to edges of . Counting can be done in two ways. One way is counting the number of perfect matchings of which is ( ), then counting its edges that is /2 . Another way is counting the number of edges of , which is , then counting the number of perfect matchings that has a special **edge**, which is .The number of perfect matchings that has a special **edge** are the same for each **edge**. Assume that = is an **edge** of then = ( \ , ). So ∗ ( ) = ∗ ( \ , ). Hence ( ) = ( \ , ).

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Let ≥ 3 be an integer, and let = ⋯ be a cycle on vertices. Let , , … , be vertex-disjoint trees, and let the root vertex of be for 1 ≤ ≤ . Denote by ( , , … , ) the unicyclic graph obtained from of by identifying the root vertex of with for 1 ≤ ≤ . Any unicyclic graph with a -cycle can be denoted by the form ( , , … , ), where | | = , ( = 1, 2, … , ) and ∑ = . By Lemma 2.3, we can repeat the **edge**-lifting transformation to the unicyclic **graphs**

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In[5], the authors introduced the notion of S− valued **graphs**, where S is a semiring. In graph theory, domination of **graphs** is the most powerful area of research for, it has several applications in other areas of sciences. It was initicted by Berge [1]. In [6], the authors have studied the vertex domination on S− valued **graphs**. In this paper we discuss the notion of **edge** domination on S− valued **graphs**.

In the following theorems, we have obtained bounds for the number of major vertices in several classes of 5-critical **graphs**, that are stronger than the existing bound given in R2. Also, we have obtained new bounds on size m in terms of order n and minor vertices. Theorem 3.1.1. Let G be a 5-critical graph. Suppose that every major vertex adjacent with a 4-vertex is adjacent with a 3-vertex.

Definition A set F of edges in a graph is called an **edge** dominating set of G if every **edge** in is adjacent to at least one **edge** in F. It can also be defined as a set F of edges G is called an **edge** dominating set of G if for every **edge** , there exists an **edge** such that and have a vertex in common. The domination number of a graph G is the minimum cardinality an **edge** dominating set of G.

Abstract. In this paper, we introduce the concepts of **edge** regular and totally **edge** regular bipolar fuzzy **graphs**. We determined necessary and sufficient condition under which **edge** regular bipolar fuzzy graph and totally **edge** regular bipolar fuzzy graph are equivalent. Some properties of **edge** regular bipolar fuzzy **graphs** are studied.

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A topological index is called degenerate if it possesses the same value for more than one graph. A set of **graphs** with the same value of a given index forms a degeneracy class. Since a topological index can be regarded as a measure of structural similarity of molecular **graphs**, the finding of information on degeneracy classes can be useful for chemical applications. There are a number of functions for characterizing degeneration of topological indices [7]. The discriminating ability of an index for a family of **graphs** can be expressed by relation

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Graph theory is a very important tool to represent many real world problems. For example, a social network may be represented as a graph where vertices represent accounts(persons, institutions, etc.) and edges represent the relation between the accounts. If the relations among the accounts are to be measured as good or bad according to the frequency of contacts among the accounts, fuzzyness should be added to representation. This and many other problems motivated to define fuzzy **graphs**. Rosenfeld (1975) introduced the notion of fuzzy **graphs** and several analogs of graph theoretic concepts such as path, cycle, connectedness etc. After that, fuzzy graph theory becomes a vast research area. Zadeh (1987) introduced the concept of fuzzy relations. The concept of a complete fuzzy **graphs** was investigated by Sunitha and Vijayakumar (2002). The concept of domination in fuzzy **graphs** was introduced by Somasundaram(1998). In this paper we introduce the new concept **semi** fuzzy **graphs**

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From the discussion in this paper, we can see that all the **edge** labels are positive and at most ⌊| G | /2 ⌋ + 3, where G is the considered graph. If | G | ≥ 7, then ⌊| G | /2 ⌋ + 3 < | G | . So, for these cases, all labelings are proper when we take modulo k, where k ≥ | G | . If 3 ≤ | G | ≤ 6 and G contains at least two cycles, then G must be 2C 3 . Corollary 5.2 shows

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G be a graph with p = | V ( G ) | vertices and q = | E ( G ) | edges. Graph labeling, where the vertices and edges are assigned real values or subsets of a set are subject to certain conditions. A detailed survey of graph labeling can be found in [2].Terms not defined here are used in the sense of Harary in [5].The concept of adjacent **edge** graceful labeling was first introduced in [18].Some results on adjacent **edge** graceful labeling of **graphs** and some non- adjacent **edge** graceful **graphs** are discussed in [18].

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In this section, we consider the **edge** energy of a graph (or the energy of the line of a graph) and obtain some of its properties. First we recall the definition of the **edge** adjacency matrix of a graph. Note that the **edge** adjacency energy of a graph is just the ordinary energy of the line graph and has been studied in detail. For instance, see [14, 21].

As a standard notation, assume that G = G(V, E) is a finite, simple and undirected graph with p vertices and q edges. Terms and terminology as in Harary [3]. Mean labeling was introduced by S.Somasundaram and R.Ponraj in [6]. In this paper, we study the **edge** odd and even mean labeling of a regular graph obtained by joining some standard **graphs**.

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This note explores a new family of **graphs** deﬁned on the set of paths of the m × n lattice. We let each of the paths of the lattice be represented by a vertex, and connect two vertices by an **edge** if the corresponding paths share more than k steps, where k is a ﬁxed parameter 0 = k = m + n. Each such graph is denoted by G(m,n,k). Two large complete subgraphs of G(m,n,k) are described for all values of m, n, and k. The size of the **edge** set is determined for n = 2, and a complicated recursive formula is given for the size of the **edge** set when k = 1.

Abstract: In this paper, degree of an egde and total degree of an **edge** in a fuzzy soft graph is introduced, **edge** regular fuzzy soft **graphs**, and totally **edge** regular fuzzy soft **graphs** are also introduced . Theorems for **edge** regular fuzzy soft **graphs** and totally **edge** regular fuzzy soft **graphs** are introduced. A necessary condition under which they are equivalent is provided. Some properties of **edge** regular fuzzy soft **graphs** and totally **edge** regular fuzzy soft **graphs** are studied.

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In this paper, we have introduced the concept of sequential join of fuzzy **graphs**, which are analogous to the concept sequential join in crisp graph theory. We study about the degree of an **edge** in fuzzy **graphs** which are obtained from three or more fuzzy **graphs** using sequential join operation. The degree of an **edge** in the sequential join of fuzzy **graphs** obtained in some particular case.

The concept of resolvability in **graphs** has previously appeared in [7, 11, 12]. Slater [11, 12] introduced this concept and motivated by its application to the placement of a minimum number of sonar detecting devices in a network so that the position of every vertex in the network can be uniquely determined in terms of its distance from the set of devices. Harary and Melter [7] dis- covered these concepts independently as well. Resolving decompositions in **graphs** were introduced and studied in [3] and further studied in [6]. Resolv- ing decompositions with prescribed properties have been studied in [5, 9, 10]. Resolving concepts were studied from the point of view of graph colorings in [1, 2]. We refer to [4] for graph theory notation and terminology not described here.

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