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On (Semi-) Edge-primality of Graphs

On (Semi-) Edge-primality of Graphs

(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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On the Zagreb Indices of Semi total Point Graphs of Some Graphs

On the Zagreb Indices of Semi total Point Graphs of Some Graphs

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.

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Vector Space semi-Cayley Graphs

Vector Space semi-Cayley Graphs

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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A comprehensive introduction to the theory of word-representable graphs

A comprehensive introduction to the theory of word-representable graphs

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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Perfect Matchings in Edge-Transitive Graphs

Perfect Matchings in Edge-Transitive Graphs

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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On the revised edge-Szeged index of graphs

On the revised edge-Szeged index of graphs

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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Edge domination on S− valued graphs

Edge domination on S− valued graphs

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.

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Edge Chromatic 5-Critical Graphs

Edge Chromatic 5-Critical 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.

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Theory of Edge Domination in Graphs-A Study

Theory of Edge Domination in Graphs-A Study

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.

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On Edge Regular Bipolar Fuzzy Graphs

On Edge Regular Bipolar Fuzzy Graphs

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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The F–Index for some Special Graphs and some Properties of the F–Index

The F–Index for some Special Graphs and some Properties of the F–Index

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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Vol 8, No 11 (2017)

Vol 8, No 11 (2017)

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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Integer-antimagic spectra of disjoint unions of cycles

Integer-antimagic spectra of disjoint unions of cycles

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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Some Adjacent Edge Graceful Graphs

Some Adjacent Edge Graceful Graphs

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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On the edge energy of some specific graphs

On the edge energy of some specific graphs

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].

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Edge Mean Labeling Of a Regular Graphs

Edge Mean Labeling Of a Regular Graphs

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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On the edge set of graphs of lattice paths

On the edge set of graphs of lattice paths

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.

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On Edge Regular Fuzzy Soft Graphs

On Edge Regular Fuzzy Soft Graphs

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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On Sequential Join of Fuzzy Graphs

On Sequential Join of Fuzzy Graphs

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.

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On resolving edge colorings in graphs

On resolving edge colorings in graphs

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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