Antimagic Labeling for Some Snake Graphs

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.

A Triangular snake Tn is obtained from a path u1u2 …un by joining ui and ui+1 to a new vertex wi for 1≤i≤n−1. A Central graph of Triangular snake C(Tn) is obtained by subdividing each edge of Tn exactly once and joining all the non adjacent vertices of Tn. In this paper the ways to construct square sum, square difference, Root Mean square, strongly Multiplicative, Even Mean and Odd Mean labeling for Triangular Snake and Central graph of Triangular Snake graphs are reported.

This research focuses on a detailed investigation of advanced fuzzy indices applied to enhanced version of triangular snake graph, designated as fuzzy triangular snake graph (mathcal {FTS}_{mathbb {N}}). Its practical applications in wireless sensor network (WSN) is also designed. We derive closed-form formulas for several fuzzy indices, including first and second fuzzy Zagreb indices, fuzzy Harmonic, Randić, Misbalance Prodeg, Nirmala, along with Sombor indices. By applying these indices to a WSN modeled with redundant triangular connectivity, we evaluate the network’s structural efficiency, robustness, and fault tolerance. The computed results reveal that the fuzzy triangular snake topology ensures balanced node connectivity, minimizes energy imbalance, and enhances the network’s ability to maintain reliable communication in the presence of node failures and uncertainties. The results confirm the applicability of the fuzzy triangular snake model in optimization of energy- and resilient WSNs, thus drawing a high degree of correlation between such contemporary graph theory, as fuzziness, and the design of a modern communication network.

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).

Sholairaju and Chithra introduced a graph which admits an edge odd graceful labeling called an edge odd graceful graph. An edge odd graceful labeling of a graph G on q edges is a bijection f : E(G) → {1,3,5, …, 2q-1} so that the induced mapping f+: V(G) → {0,1,2, …,2q - 1} given by f+(x) = Σxy∈E(G) f(xy) (mod 2q) is injective. A triangular snake C3m is a graph obtained from a path u1u2u3 … um+1 by joining every ui and ui+1 to a new vertex vi. A quadrilateral snake C4m is a graph obtained from vertices u1, u2, u3, …, um+1 by joining every ui and ui+1 to two vertices vi and wi. In this pa r we study edge odd graceful labelings of triangular snake C3m and quadrilateral snake C4m. We find a labeling of C3m and a labeling of C4m. This implies that graphs C3m and C4m are odd graceful, and we have new classes of graphs that are odd graceful.

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.

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.

Let G be a graph with p vertices and q edges and A = {1, 3, ..., q} if q is odd or A = {1, 3, ..., q + 1} if q is even. A graph G is said to admit an odd vertex equitable even labeling if there exists a vertex labeling f : (G) → A that induces an edge labeling f∗ defined by f∗(uv) = f(u) + f(v) for all edges uv such that for all a and b in A, |vf(a) − vf(b)| ≤ 1 and the induced edge labels are 2, 4, ..., 2q where vf(a) be the number of vertices v with f(v) = a for a ∈ A. A graph that admits an odd vertex equitable even labeling is called an odd vertex equitable even graph. Here, we prove that the subdivision of double triangular snake (S(D(Tn))), subdivision of double quadrilateral snake (S(D(Qn))), DA(Qm) ⊙ nK1 and DA(Tm) ⊙ nK1 are odd vertex equitable even graphs.

Let G be a (p,q) graph. An injective map f : E(G) → {±1,±2,...,±q} is said to be an edge pair sum labeling if the induced vertex function f*: V (G) → Z - {0} defi ned by f*(v) = ΣP∈Ev f (e) is one-one where Ev denotes the set of edges in G that are incident with a vertex v and f*(V (G)) is either of the form {±k1,±k2,...,±kp/2} or {±k1,±k2,...,±k(p-1)/2} U {±k(p+1)/2} according as p is even or odd. A graph with an edge pair sum labeling is called an edge pair sum graph. In this paper we prove that the graphs GL(n), double triangular snake D(Tn), Wn, Fln, and admit edge pair sum labeling.

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.

In a graph , define Pell labeling is a map : () {0,1, , -1} with an induced function * : () defined by * () = () + 2() for every () are all distinct where , 0. In addition to this a graph which admits Pell labeling concept is known as Pell graph.In this paper, the Pell labeling concepts applied for the following graphs such as splitting of Star, cycle with parallel chords, alternate double triangular Snake, ladder, Triangular ladder, diagonal ladder, shadow and splitting of path, bistar, subdivision of bistar, prism, , 2 and friendship graphs are studied.Our analysis contributes to the understanding of Pell labeling across a broad spectrum of graph configurations, highlighting its applicability and the unique characteristics of each considered graph family.

<p>We prove that closed helm CHn, web graph Wbn, flower graph Fln, double<br /> triangular snake DTn and gear graph Gn admit product cordial labeling.</p>

Dominator coloring is an important area of graph theory with applications and coloring methods often overlook the dominance relationships between vertex color classes, leading to inefficiencies. This paper introduces dominator sum coloring, a novel hybrid approach combining dominator coloring and sum coloring to address these inefficiencies. The goal of this study is to assign labels to vertices that minimize the sum of labels while ensuring each vertex dominates at least one-color class, thereby reducing the number of color classes needed. We apply dominator sum coloring to triangular snake graph families, including triangular snake T<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</inf>, double triangular snake DT<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</inf>, triple triangular snake TT<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</inf>. Through algorithmic techniques, we compute the dominator sum chromatic number, χ<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ds</inf>(G) and the chromatic sum S<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ds</inf>(G) graph. for these graphs. Our results show that the dominator sum chromatic number significantly reduces the number of color classes required compared to traditional dominator coloring. Additionally, the chromatic sum is minimized, leading to a more efficient allocation of vertex labels. This paper compares dominator sum coloring within dominator coloring methods, focusing on computational efficiency and real-world applications. The proposed approach outperforms conventional methods in terms of both computational complexity and optimization. It offers an efficient solution for managing dominance relationships in graphs, with practical applications in wireless network design, social network analysis, and resource allocation, where optimizing dominance can improve overall performance.

The quantitative specifications which are used to narrate the atomic topology of graphs are usually entitled as topological indices in theoretical Chemistry. The features of physical and chemical properties like melting point, entropy, boiling point, energy generation and enthalpy of vaporization of chemical compounds can be estimated by means of these topological indices. The field of Graph theory has a remarkable application of linking certain graphs with many types of topological indices. In this paper we compute first and second Zagreb indices, Randi´c index, sum-connectivity index, harmonic index, inverse sum indeg index, modified first and second Zagreb indices and first and second hyper Zagreb indices of Triangular snake graph and alternate Triangular snake graph.