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.

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 =

1.1. Signed **Graphs** and Their IASLs. A signed graph (see [15]), 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.

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 [7] for denitions of these families).

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 .

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.

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tress have **prime** labeling, which is not settled till today. The **prime** labeling for planner grid is investigated by M. Sundaram [7]. S.K. Vaidhya and K.K. Kanmani have proved that the **prime** labeling for **some** cycle related **graphs** [9]. S. Meena and K. Vaithilingam, **Prime** Labeling for **some** Helm related **graphs** [6].

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.

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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.

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**.

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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 [4].The edge labels are the actual images under function f:E(G)→{1,2,….|E|} Deretsky,Lee and Mitchen [4] 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

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in [11]. The concept of neighborhood-**prime** labeling of graph was introduced by Patel et al. [10]. Lawrence et al. introduced the notation of k-neighborhood-**prime** labeling of graph in [8]. Lau et al was introduced a variant of **prime** graph labeling of graph in [6]. In [7], Lau et al. introduced SD-**prime** cordial labeling and they discussed SD-**prime** cordial labeling for **some** standard **graphs**. In [9], Lourdusamy et al. investigated **some** **new** construction of SD-**prime** cordial graph. In [3], 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 [1], 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

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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.

The concept of **prime** cordial labeling was introduced by Sundaram [5] et al. and in the same paper they have investigated several **results** on **prime** cordial labeling. Vaidya and Vihol [6] have also discussed **prime** cordial labeling in the context of graph operations while in [7] the same authors have discussed **prime** cordial labeling for **some** cycle related **graphs**. Vaidya and Shah [8] have investigated many **results** on this concept. In the present

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.

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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 [4] . 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.

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and N.Kumaravel[5] 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[6] discussed edge domination in fuzzy **graphs**. A.Nagoorgani and M.Baskar Ahamed[7] discussed order and size in fuzzy graph. A.Nagoorgani and J.Malarvizhi [8] 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**.

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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]

By a graph, we mean a finite, undirected graph without loops and multiple edges, for terms not defined here, we refer to Harary [4]. For standard terminology and notations related to number theory we refer to Burton [2] and graph labeling, we refer to Gallian [3]. The notion of **prime** labeling for **graphs** originated with Roger Entringer and was introduced in a paper by Tout et al. [8] in the early 1980s and since then it is an active field of research for many scholars. Patel et al.[6] 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 [1]. In [9], 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

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of vertices and the edges labelled with x. A graph with a k-total **prime** cordial labeling is called k-total **prime** cordial graph. In this paper we investigate the 4-total **prime** cordial labeling of **some** **graphs** like Prism, Helm, Dumbbell graph, Sun flower graph.