# Top PDF 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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### Adjacent Edge Graceful Graphs

All graphs in this paper are finite, simple and undirected graphs. Let ( p , q ) 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 [1].Terms not defined here are used in the sense of Harary in [3].The concept of edge graceful labeling was first introduced in [2] and the concept of strong edge graceful labeling was introduced in [4]. Some results on strong edge graceful labeling of graphs are discussed in [4]. In this paper, we introduced a new edge graceful labeling. We use the following definitions in the subsequent sections.
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### Adjacent Vertex Distinguishing Proper Edge Colorings of Bicyclic Graphs

the avd-colorings of graphs, see [3–5, 7–9, 11, 13–16, 18]. A bicyclic graph is a connected graph in which the num- ber of edges equals the number of vertices plus one. In this paper, we investigate the avd-coloring of bicyclic graphs and show that χ a ′ (G) ≤ ∆(G) + 1 for bicyclic graphs G. This implies that Conjecture 1 holds for all bicyclic graphs.

### Adjacent Vertex Distinguishing Edge colorings of the Lexicographic Product of Special Graphs

chromatic number of coloring, For special H , the exact value of the adjacent vertex distinguishing edge-coloring of G [ H ] is obtained. In this paper, we prove that the chromatic number of adjacent vertex distinguishing edge-coloring of lexicographic product G [ H ] for any two graphs G and H is equal to the graph class

### Snakes and Caterpillars in Graceful Graphs

m may change from edge to edge; they called this replacement, supersubdivision. More general results about super subdivisions can be found in [2], [3], and [18]. In Section 3 we analyze this concept in more detail. There, we introduce a replacement theorem that allows us to replace, within a graceful labeled graph, some specific labeled subgraphs by some analogous graphs.

### Adjacent Vertex Distinguishing (Avd) Edge Colouring of Permutation Graphs

The tables which are constructed in the theorem1 and theorem2 are not unique. There are many ways to construct the tables but in every way the required minimum no. of colours for avd edge colouring in the table is same as the theorems. Also the tables are constructed in the theorem 2 are symmetric. Any one can work on algorithm of avd edge colouring of any arbitrary permutation graphs.

### On \(k\)-graceful labeling of pendant edge extension of complete bipartite graphs

again have some more cases to consider, depending on the values of i, and j. When 1 6 i 6 r, or p(n − 1) + r + 1 6 i 6 min{(p + 1)(n − 1) + r, m}, then similar to Case(1), and Case(2), we can easily observe that q is not in A 1 , A 2 . The remaining case that we need to consider

### Pentagonal Graceful Labeling of Caterpillar Graphs

Abstract - A graph G = (V, E) with p vertices and q edges is said to admit pentagonal graceful labeling if its vertices can be labeled by non negative integers such that the induced edge labels obtained by the absolute difference of the labels of end vertices are the first 𝒒 pentagonal numbers. A graph 𝑮 which admits pentagonal graceful labeling is called a pentagonal graceful graph. In this paper, we prove that Caterpillar is a pentagonal graceful graph.

### Odd Graceful Labeling of the Revised Friendship Graphs

A graph G of size q is odd-graceful, if there is an injection  from V(G) to {0, 1, 2, …, 2q-1} such that, when each edge xy is assigned the label or weight |  (x) -  (y)|, the resulting edge labels are {1, 3, 5, …, 2q-1}. This definition was introduced in 1991 by Gnanajothi [1] who proved that the class of odd graceful graphs lies between the class of graphs with α-labelings and the class of bipartite graphs. Gnanajothi [1] proved that every cycle C n is odd graceful if n is even. It is
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### Gracefulness of Some Super Graphs of KC4 -Snake

Proof: Let H be the kC 4 -snake with k blocks, so the number of vertices of H is 3 k + 1 and the number of edges of H is 4 . Let G be the super graph of H such that G is the k complete m-points projection on the disjoint vertices of kC 4 -snake. Let N = m + 3 k + 1 be the number of vertices of G and M = 2 mk + 4 k be the number of edges of G. [Refer Figure 2.5]. To prove G is graceful it is enough to prove that the M edges o f G having the edge values as { M , M − 1 , M − 2 , ..., 3 , 2 , 1 } . Name the k + 1 adjoint vertices by
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### Vol 8, No 10 (2017)

(graphs obtained by joining a single pendent edge to each vertex of C n ) if and only if n is even ; the disjoint union of copies of C 4 ; the one-point union of copies of C 4 ; C n × K 2 if and only if n is even; caterpillars; rooted trees of height 2; the graphs obtained from P n (n > 3) by adding exactly two leaves at each vertex of degree 2 of P n ; the graphs obtained from P n × P 2 by deleting an edge that joins to end points of the P n paths ; the graphs obtained from a star by adjoining to each end vertex the path P 3 or by adjoining to each end vertex the path P 4 . She conjectures that all trees are odd-graceful and proves the conjecture for all trees with order up to 10. Barrientos [2] has extended this to trees of order up to 12. For details in the progress made so far in this area one can refer to the latest Survey on graph labeling problems due to Gallian [3]. Here we are inspired by the works of Solairaju [4] and Moussa and Badr [5] and give odd graceful labeling to the graphs : Bistar, K 1 , n : 2 ,
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### Missing Numbers in K Graceful Graphs

, Then Rosa [10] called the mapping the β-labeling (valuation) of a graph G, Golomb [4] subsequently called such labeling to be graceful labeling and the graph is called a graceful graph, while is called an induced edge’s graceful labeling. -graceful labeling is the generalization of graceful labeling that introduced by Slater [11] in 1982 and by Maheo and Thuillier [8] in 1982.

### EDGE DOMINATION IN SOME BRICK PRODUCT GRAPHS

By a graph G = (V, E) we mean a finite undirected graph without loops and multiple edges. A subset S of V is called a dominating set of G if every vertex not in S is adjacent to some vertex in S. The domination number of G denoted by γ(G) is the minimal cardinality taken over all dominating sets of G. A subset F of E is called an edge dominating set if each edge in E is either in F or is adjacent to an edge in F . An edge dominating set F is called minimal if no proper subset of F is an edge dominating set. The edge domination number of G denoted by γ 0 (G) is the minimum cardinality taken over all edge dominating sets of G.
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### On the edge and total GA indices of some graphs

In [5], the edge version of geometric-arithmetic index introduced based on the end-vertex degrees of edges in a line graph of G which is a graph such that each vertex of L(G ) represents an edge of G; and two vertices of L(G) are adjacent if and only if their corresponding edges share a common endpoint in G, as follows

### Some Bounds of Rainbow Edge Domination in Graphs

and means f (e) = 0 and since f : E(G) → {0,1,2} is –function, then any edge must be adjacent to edge h , i.e.,f (h) = 2 and in the function g : E(G) → {1,2}, if then g( h ) = {1,2}. Hence g : E(G) → {1,2} is a 2-rainbow edge domination function in G with the weight W(g) that means

### Constructing Graceful Graphs with Caterpillars

A graceful labeling of a graph G of size n is an injective assignment of integers from {0, 1, . . . , n} to the vertices of G, such that when each edge of G has assigned a weight, given by the absolute difference of the labels of its end vertices, the set of weights is {1, 2, . . . , n}. If a graceful labeling f of a bipartite graph G assigns the smaller labels to one of the two stable sets of G, then f is called an α-labeling and G is said to be an α-graph. A tree is a caterpillar if the deletion of all its leaves results in a path. In this work we study graceful labelings of the disjoint union of a cycle and a caterpillar. We present necessary conditions for this union to be graceful and, in the case where the cycle has even size, to be an α- graph. In addition, we present a new family of graceful trees constructed using α-labeled caterpillars.
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### Odd Graceful Labeling of Some New Type of Graphs

The study of graph labeling and graceful graphs was introduced by Rosa[4].(Gnanajothi R.B,1991)[5 ] introduced the concept of odd graceful graphs and she has proved many results. Kathiresan K.M.,2008[6] has discussed odd gracefulness of ladders and graphs obtained fromthem by subdividing each step exactly once. Vaidya.S.K. e-tal 2010[7 ]proved the odd gracefulness of joining of even cycle with path and cycle sharing a common edge. Vaidya S.K. e-tal 2013[ 8] proved the odd gracefulness of splitting graph and the shadow graph of bister. Barrientos Christian 2009[ 9] discussed the odd gracefulness of Trees of Diameter 5.For detailed survey on graph labeling and related results we refer to Gallian J.A.,2015[10 ]
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### Super Edge Graceful and Even Edge Graceful Labeling of Cayley Digraphs

Rosa [2] introduced the notion of graceful labeling. Bloom and Hsu [3],[4],[5] extended the concept of graceful labeling for digraphs also. Lo [6] introduced the idea of edge graceful graphs. Lee [7] further developed the concept for more graphs. Lee et al [8] defined super edge graceful labeling for some undirected graphs. Also Gayathri et al [9] discussed the even edge graceful labeling for undirected graphs. The Cayley graph that presents the exact structure of a group and this definition is introduced by Cayley [10]. Thirusangu et al [11] defined labeling concepts for some classes of Cayley digraphs. Further many authors started various labeling technologies on the Cayley digraphs [12],[13]. We now extended super edge graceful and the even edge graceful labeling to directed graphs and applied both the methodologies on the Cayley digraphs.
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### Super Edge-antimagic Graceful labeling of Graphs

We look at a computer network as a connected undirected graph. A network designer may want to know which edges in the network are most important. If these edges are removed from the network, there will be a great decrease in its performance. Such edges are called the most vital edges in a network [5, 6, 12]. However, they are only concerned with the effect of the maximum flow or the shortest path in the network. We can consider the effect of a minimum spanning tree in the network. Suppose that G = (V, E) is a weighted graph with a weight w(e) assigned to every edge e in G. In the weighted graph G, the weight of a spanning tree T, w(T ) is defined to be P w(e)
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### Edge even graceful labeling of some graphs

mod (2k), is an injective function, where k = max(p, q). The graph that admits an edge even graceful labeling is called an edge even graceful graph. In Fig. 1 , we present an edge even graceful labeling of the Peterson graph and the complete graph K 5 .