Kth FIBONACCI PRIME LABELING OF SNAKE GRAPHS

An injective function f : V (G) → {T0, T1, T2, . . . , Tn}, where n = |V (G)| − 1, is said to be a Tribonacci cordial labeling if the induced function f ∗ E(G) →{0, 1} defined by f∗(uv) = (f(u) + f(v)) mod 2 satisfies the condition |ef (0) − ef (1)| ≤ 1 where ef (0) is the number of edges with label 0 and ef (1) is the number of edges with label 1. A graph that admits a tribonacci cordial labeling is called a Tribonacci cordial graph. In this paper, we determined the Tribonacci Cordial Labeling of Triangular Snake Graph T Sn, Double Triangular Snake Graph D(T Sn), Quadrilateral Snake Graph QSn, Double Quadrilateral Snake Graph D(QSn), and Cycle Quadrilateral Snake Graph C(QSn).

The product cordial labeling is a variant of cordial labeling. Here we investigate product cordial labelings for alternate triangular snake and alternate quadrilateral snake graphs.

A variation of the channel-assignment problem is naturally modeled by L(2,1)-labelings of graphs. An L(2,1)-labeling of a graph G is an assignment of labels from {0,1,...,/spl lambda/} to the vertices of G such that vertices at distance two get different labels and adjacent vertices get labels that are at least two apart and the /spl lambda/-number /spl lambda/(G) of G is the minimum value /spl lambda/ such that G admits an L(2,1)-labeling. The /spl Delta//sup 2/-conjecture asserts that for any graph G its /spl lambda/-number is at most the square of its largest degree. In this paper it is shown that the conjecture holds for graphs that are direct or strong products of nontrivial graphs. Explicit labelings of such graphs are also constructed.

Let G = (V,E) be a simple graph. L(2, 1)−labeling defined as a functionf : V (G) → N0 such that, x and y are two adjacent vertices in V, then if x andy are adjacent to each other, |f(y) − f(x)| ≥ 2 and if x and y have the distance 2,|f(y) − f(x)| ≥ 1. The L(2, 1)-labeling number of G, called λ2,1(G), is the smallestnumbermof G. In this paper, we will further discuss the L(2, 1)-labeling of mongoliantent, lobster, triangular snake, and kayak paddle.Keywords: L(2,1)-Labeling, mongolian tent, lobster, triangular snake, kayak paddle.

One important problem in graph theory is graph coloring or graph labeling.Labeling problem is a well-studied problem due to its wide applications, especially in frequency assignment in(mobile) communication systems, coding theory, x-ray crystallography, radar, circuit design, etc.For two non-negative integers, labeling of a graph is a function from the node set to the set of non-negative integers.For any two distinct vertices and of , a radio labeling is an injective function : 0 such that , | | 1 where is the diameter of .The radio number of , is the maximum value of taken overall radio labelings of .This paper determines the radio number of double triangular snake graphs.

An edge odd graceful labeling of graph G with q number of edges is a bijection f : E(G) → {1,3,5, …,2q - 1} so that induced mapping f+ : V(G) → {0, 1,2, … ,2q - 1} given by f+(x) = ExyEE(G) f(xy) (mod 2q) is injective. A triangular snake net graph (m - 1)C3m is graph obtained from a path u1 u2 u3 … um + 1 by joining every ui and ui + 1 to a new vertex vi with 1 ≤ i ≤ m and joining every vi to vi+1 with 1 ≤ i ≤ m - 1. A quadrilateral snake net graph C4,4m is a graph obtained from vertices u1, u2, u3, …, u2m + 1 by joining every ui and ui + 1 to two vertices vi and v2i-1with 1 ≤ i ≤ 2m, and joining every v4i - 2 and v4i to a new vertex w2i and joining every v4i - 3 and v4i-1 to a new vertex w2i - 1 with 1 ≤ i ≤ m. An alternate quadrilateral snake net A(C4,4m) is graph obtained from vertices u1, u2, u3, … , u3m by joining every u3i and u3i + 1 with 1 ≤ i ≤ m - 1, and joining u3i - 2 and u3i - 1 to two new vertices v4i-3 and v4i-2, joining u3i-1 and u3i to two new vertices v4i-1 and v4i, joining v4i-3 and v4i-1 to a new vertex w2i-1, and joining v4i-2 and v4i to two a new vertex w2i with 1 ≤ i ≤ m. A quadrilateral pleated snake graph C42 is a graph obtained from vertices u0, u, and v by joining every u0 and u to new vertices ui and joining every u0 and v to new vertices vi with 1 ≤ i ≤ 2 m + 2. In this paper, we study edge odd graceful labeling for the following families of snake graphs such as triangular snake net, quadrilateral snake net, alternate quadrilateral snake net, and quadrilateral pleated snake.

Klein cordial trees and odd cyclic cordial friendship graphs

In graph theory many useful mathematical disciplines, but the extremely useful one is graph labeling. A graph labeling is used in many applications in various fields like X-ray crystallography, computer science, data base management, coding theory, astronomy and physics. A labeling of edges and vertices of a simple graph G(V, E) by a mapping ¥ : V(G) ∪ E(G) → {1, 2, 3, … , ℜ} provided that any two different edges have distinct weights is called an edge irregular total ℜ -labeling. If ℜ is minimum and G admits an edge irregular total ℜ-labeling, then it is called the total edge irregularity strength (TEIS) denoted by tes(G). In this paper, we have introduced three definitions: the quintet snake graph QS n , the double quintet snake graph D(QS n) and the m–multiple quintet snake graph M(QS n). Moreover, the exact value of TEIS for the new graphs are investigated.

A graph with a vertex set is said to be a prime graph if there exists a bijective mapping , where denotes the number of vertices in , such that for any two adjacent vertices and in have . Tensor Product graph is a way to combine (compose) two graphs into one larger and more complex graph. The result is a new graph that reflects the connection properties of the two original graphs, but in a very specific and more complex way than other graph operations. Therefore, this research aims to determine whether there is prime labeling in the class of graphs resulting from the Tensor Product of the path graph and the cycle graph . The research employed analytical and exploratory methods with a trial-and-error strategy to determine the labeling that possesses a prime property. The results of this study prove that two classes of the Tensor Product graph for , and graph , for are prime graph. This finding expands the results on classes of graphs that admit prime labeling and provides a basis for further research on graph labeling in other graph operations

A graph G is an ordered pair of sets G(V, E), where V is a set of vertices and the elements of set E are usually called edges. The concept of graph labeling has recently gained a lot of popularity in the area of graph theory. his popularity is due not only to the mathematical challenges of graph labelings but also to the wide range of applications that graph labeling offer to other brances of science, for instance, x-ray, cryptography, coding theory, circuit design and communication network design. A graph labeling we mean an assignment of integers to elements of a graph such as vertex, edge, and both. Local antimagic of a graph was motivated by Arumugam et al.. Thus, we initiate to developed the concept of local antimagic vertex coloring for subdevided graphs. A graph G called sub-devided if the graph G be the graph obtained by inserting a vertex to each edge of the graph G. Definition the concept local antimagic vertex coloring is f : E(G) → {1, 2, 3…, |E(G)|} if for any two adjacent vertices a 1 and a 2, w(a 1) = w(a 2), where for v ∈ G, , where E(a) and V(a) are respectively the set of edges incident to a. The local antimagic vertex labeling induces a proper vertex coloring of graph G if each vertex a is assigned the color w(a). The minimum colors needed to coloring the vertices in graph G called local antimagic vertex chromatic number, denoted by χla (G). In this paper we study the local antimagic vertex coloring of sub-devided some special graph as follows SFn,m , SSn,m , SWn,m and SFn,m .

The process of assigning the binary numbers 0 and 1 to the edges of a graph \(G = (V, E)\) through evaluating functions defined on the vertex set V and the edge set E of G using the concepts of prime and Narayana numbers by satisfying cordiality on the edges. This process on the graph G is known as Narayana prime cordial labeling of G and this graph G is called Narayana prime cordial graph. In this article, we compute the Narayana prime cordial labeling of Web graphs and Flower graphs.

L ( 2 , 1 ) -labeling is a graph coloring model inspired by a frequency assignment in telecommunication. It asks for such a labeling of vertices with nonnegative integers that adjacent vertices get labels that differ by at least 2 and vertices in distance 2 get different labels. It is known that for any k ⩾ 4 it is NP-complete to determine if a graph has a L ( 2 , 1 ) -labeling with no label greater than k. In this paper we present a new bound on complexity of an algorithm for finding an optimal L ( 2 , 1 ) -labeling (i.e. an L ( 2 , 1 ) -labeling in which the largest label is the least possible). We improve the upper complexity bound of the algorithm from O ⁎ ( 3.5616 n ) to O ⁎ ( 3.2361 n ) . Moreover, we establish a lower complexity bound of the presented algorithm, which is Ω ⁎ ( 3.0739 n ) .

Let G be a graph. A bijection f:V→ {1,2,…..|V|} is called a prime labeling [3] if for each edge e=uv in E, we have GCD{ f(u),f(v)}=1. A graph that admits a prime labeling is said to be a prime graph. In this paper we show that bull graph admits Prime labeling in the context of variety graph operations namely duplication of vertex, fusion of vertices and Switching in Bull graph.

An $$L(j,k)$$L(j,k)-labeling of a graph is a vertex labeling such that the difference of the labels of any two adjacent vertices is at least $$j$$j and that of any two vertices of distance $$2$$2 is at least $$k$$k. The minimum span of all $$L(j,k)$$L(j,k)-labelings of $$G$$G is denoted by $$\lambda _k^j(G)$$?kj(G). Lin and Lam (Discret Math 308:3805---3815, 2008) provided an upper bound of $$\lambda _1^2(K_m \times C_n)$$?12(Km×Cn) when $$K_m \times C_n$$Km×Cn is the direct product of a complete graph $$K_m$$Km and a cycle $$C_n$$Cn. And they found the exact value of $$\lambda _1^2(K_m \times C_n)$$?12(Km×Cn) for some $$m$$m and $$n$$n. In this paper, we obtain an upper bound and a lower bound of $$\lambda _1^1(K_m \times C_n)$$?11(Km×Cn). As a consequence we compute $$\lambda _1^1(K_m \times C_n)$$?11(Km×Cn) when $$n$$n is even or $$n\ge 4m+1$$n?4m+1.

Let G=(V,E) be a simple graph. A vertex labeling f:V(G)→{1,2,⋯,k} is defined to be a local inclusive (respectively, non-inclusive) d-distance vertex irregular labeling of a graph G if for any two adjacent vertices x,y∈V(G) their weights are distinct, where the weight of a vertex x∈V(G) is the sum of all labels of vertices whose distance from x is at most d (respectively, at most d but at least 1). The minimum k for which there exists a local inclusive (respectively, non-inclusive) d-distance vertex irregular labeling of G is called the local inclusive (respectively, non-inclusive) d-distance vertex irregularity strength of G. In this paper, we present several basic results on the local inclusive d-distance vertex irregularity strength for d=1 and determine the precise values of the corresponding graph invariant for certain families of graphs.