# Top PDF Some New Results on Prime Graphs ### Some New Results on Prime Graphs

Labelings of Graphs,” In: J. Alvi, G. Chartrand, O. Oel- lerman, A. Schwenk, Eds., Graph Theory, Combinatorics and Applications: Proceedings of the 6th International Conference Theory and Applications of Graphs, Wiley, New York, 1991, pp. 359-369. ### New Results on Vertex Prime Graphs

vertex prime. They have further proved that a graph with exactly 2 components, one of which is not an odd cycle has a vertex prime labeling and a 2 – regular graph with atleast two odd cycles does not have a vertex prime labeling. They have conjectured that a 2 – regular graph has a vertex prime labeling if and only if it does not have two odd cycles. Let G = ### SOME NEW RESULTS ON INTEGER ADDITIVE SET-VALUED SIGNED GRAPHS

1.1. Signed Graphs and Their IASLs. A signed graph (see ), denoted by Σ(G, σ), is a graph G(V, E) together with a function σ : E(G) → {+, −} that assigns a sign, either + or −, to each ordinary edge in G. The function σ is called the signature or sign function of Σ, which is defined on all edges except half edges and is required to be positive on free loops. ### New results on word-representable graphs

The notion of word-representable graphs has its roots in the study of the celebrated Perkins semi- group [10, 14]. These graphs possess many attractive properties (e.g. a maximum clique in such graphs can be found in polynomial time), and they provide a common generalization of several important graph families, such as circle graphs, comparability graphs, 3-colorable graphs, graphs of vertex degree at most 3 (see  for denitions of these families). ### A Study on Prime Labeling of Some Special Graphs M. Ramya, C. Nandhini

Let and be two distinct vertices of a graph new graph is constructing by Fusing(identifying) two vertices and by a single vertex in such that every edge which was incident with either (or) in now incident with in . ### Bounds on the number of closed walks in a graph and its applications

Using graph-theoretical techniques, we establish an inequality regarding the number of walks and closed walks in a graph. This inequality yields several upper bounds for the number of closed walks in a graph in terms of the number of vertices, number of edges, maximum degree, degree sequence, and the Zagreb indices of the graph. As applications, we also present some new upper bounds on the Estrada index for general graphs, bipartite graphs, trees and planar graphs, some of which improve the known results obtained by using the algebraic techniques. ### Some New Outcomes on Prime Labeling

tress have prime labeling, which is not settled till today. The prime labeling for planner grid is investigated by M. Sundaram . S.K. Vaidhya and K.K. Kanmani have proved that the prime labeling for some cycle related graphs . S. Meena and K. Vaithilingam, Prime Labeling for some Helm related graphs . ### SQUARE DIFFERENCE PRIME LABELING –MORE RESULTS ON CYCLE RELATED GRAPHS

Square difference prime labeling of a graph is the labeling of the vertices with {0,1,2- --,p-1} and the edges with absolute difference of the squares of the labels of the incident vertices .The greatest common incidence number of a vertex (gcin) of degree greater than one is defined as the greatest common divisor of the labels of the incident edges. If the gcin of each vertex of degree greater than one is one, then the graph admits square difference prime labeling. Here we investigate, duplicating an edge in a cycle, duplicating a vertex by an edge in a cycle, duplicating an edge by a vertex in cycle, switching a vertex in cycle, strong duplicate graph of cycle, crown graph, prism graph and two copies of cycle sharing a common vertex, for square difference prime labeling. ### Two New Results on Permutation Graphs

Abstract. The permutation graph is a very important subclass of intersection graphs. This graph class is used to solve many real life problems. In this article, an alternative proof is given to show that every path is a permutation graph. Also, it is proved that a lobstar is a permutation graph. ### Some Results On Prime Labeling Of Graphs

A graph G with n vertices is said to admit prime labeling if its vertices can be labeled with distinct positive integers not exceeding n such that the labels of each pair of adjacent vertices are relatively prime. A graph G which admits prime labeling is called a prime graph. In this paper we have proved that some classes of graphs such as the flower pot, coconut tree, umbrella graph, shell graph, carona of a shell graph, carona of a wheel graph, carona of a gear graph, butterfly graph, Two copies of cycle C n having a common vertex and carona of a alternative triangular snake are prime graphs. ### Some Vertex Prime Graphs And A New Type Of Graph Labeling

All graphs considered here are simple ,finite,connected ,undirected.A graph G(V,E) has vertex prime labeling if it’s edges can be labeled with distinct integers 1,2,3..|E| ie a function f:E→{1,2,..|E|} defined such that for each vertex with degree at least 2 the the greatest common divisor of the labels on it’s incident edges is 1 .The edge labels are the actual images under function f:E(G)→{1,2,….|E|} Deretsky,Lee and Mitchen  shows that the forests,all connected graphs,C 2k UC n ,5C 2m have vertex prime labeling.The graph with exactly two components one of ### Abstract. In this paper, we investigate the SD-Prime cordial labeling of Pl n

in . The concept of neighborhood-prime labeling of graph was introduced by Patel et al. . Lawrence et al. introduced the notation of k-neighborhood-prime labeling of graph in . Lau et al was introduced a variant of prime graph labeling of graph in . In , Lau et al. introduced SD-prime cordial labeling and they discussed SD-prime cordial labeling for some standard graphs. In , Lourdusamy et al. investigated some new construction of SD-prime cordial graph. In , Delman et.al., introduced the concept of k-SD-prime cordial labeling of graph and discussed k-SD-prime cordial labeling of some standard graphs. In , Babujee defined a class of planar graph as graph obtained by removing certain edges from the corresponding complete graph. The class of planar graph so obtained is denoted by Pl n . Here we discuss the SD-Prime cordial labeling of Pl n ### SUM SQUARE PRIME LABELING OF SOME SNAKE GRAPHS

Here we proved the results for some snake graphs. This topic is open for all other researchers to prove more snake graphs admit sum square prime labeling. Graph labeling as a whole has applications in various scientific and engineering problem. So surely sum square prime labeling is also useful to solve various application problems. ### Prime Cordial Labeling of Some Graphs

The concept of prime cordial labeling was introduced by Sundaram  et al. and in the same paper they have investigated several results on prime cordial labeling. Vaidya and Vihol  have also discussed prime cordial labeling in the context of graph operations while in  the same authors have discussed prime cordial labeling for some cycle related graphs. Vaidya and Shah  have investigated many results on this concept. In the present ### Results on Relatively Prime Dominating Sets in Graphs

Abstract. In this paper we introduce relatively prime dominating set of a graph G. Let G be a non–trivial graph. A set S ⊆ V is said to be relatively prime dominating set if it is a dominating set with at least two elements and for every pair of vertices u and v in S such that (deg u, deg v) = 1. The minimum cardinality of a relatively prime dominating set is called relatively prime domination number and it is denoted by γ rpd (G) . If there is no such pair exist then γ rpd (G) = 0. We characterize connected unicyclic graphs with γ rpd (G) =2 and also we prove that γ rpd (K m, n ) = 2 iff (m, n) = 1 and γ rpd ( P n ) = 2 for n ≥ 4. ### Some New Results on Sum Divisor Cordial Graphs

By a graph, we mean a finite undirected graph without loops or multiple edges. For standard terminology and notations related to graph theory we refer  . A labeling of graph is a map that carries the graph elements to the set of numbers, usually to the set of non-negative or positive integers. If the domain is the set of edges, then we speak about edge labeling. If the labels are assigned to both vertices and edges, then the labeling is called total labeling. Cordial labeling is extended to divisor cordial labeling, prime cordial labeling, total cordial labeling, Fibonacci cordial labeling etc. The total cordial labeling concept is further extended to edge magic total labeling, edge trimagic total labeling, 3-equitable and total magic cordial labeling etc. ### SOME RESULTS ON BIPOLAR FUZZY GRAPHS

and N.Kumaravel introduced the concept of edge degree, total edge degree and discussed about the degree of an edge in some fuzzy graphs. S.Arumugam and S.Velammal discussed edge domination in fuzzy graphs. A.Nagoorgani and M.Baskar Ahamed discussed order and size in fuzzy graph. A.Nagoorgani and J.Malarvizhi  discussed properties of 𝜇 complement of a fuzzy graph. In this paper we introduce strongly irregular bipolar fuzzy graph and strongly total irregular bipolar fuzzy graph. We provide some results on strongly irregular bipolar fuzzy graphs and strongly total irregular bipolar fuzzy graphs. ### Some Results On Divisor Cordial Labeling Of Graphs

In the above labeling, see that the consecutive adjacent vertices having the labels even numbers and consecutive adjacent vertices having labels odd and even numbers contribute 1 to e[r] ### K-Neighborhood-Prime Labeling Of Graphs

By a graph, we mean a finite, undirected graph without loops and multiple edges, for terms not defined here, we refer to Harary . For standard terminology and notations related to number theory we refer to Burton  and graph labeling, we refer to Gallian . The notion of prime labeling for graphs originated with Roger Entringer and was introduced in a paper by Tout et al.  in the early 1980s and since then it is an active field of research for many scholars. Patel et al. introduce the notion of neighborhood-prime labeling of graph and they present the neighborhood-prime labeling of various graphs in [6,7]. Ananthavalli et al. present the neighborhood-prime labeling of some special graphs in . In , Vaidya et al. introduce the concept of k-prime labeling of graphs. Lawrence et al. introduce the notation of k-neighborhood- prime labeling and they present the neighborhood-prime labeling of G ∗ B B, where B is the book with triangular and rectangle pages, G 