New classes of graphs with edge $ \; \delta- $ graceful labeling

Graceful graphs were first studied by Rosa in 1966. The Kotzig-Ringel graceful tree conjecture states that every tree has a graceful labeling. Aldred and McKay and others have used computer programs to show that trees of order up to 35 are graceful. Bagga et al. investigated algorithms for generating all graceful labelings of certain known classes of graceful graphs, including paths, cycles, and certain other classes of unicyclic graphs. The data generated by such algorithms has led to the discovery of new properties of such graceful labelings. In this paper we present a survey of graceful graph labeling algorithms and related complexity issues.DOI : http://dx.doi.org/10.22342/jims.0.0.14.1-9

In this paper, we studied the Incident Binary Equivalent Decimal Edge Graceful Labeling (IBEDE) [1] and Strong Incident Binary Equivalent Decimal Edge Graceful Labeling (SIBEDE) [1] for Wheel Graph W1,n and Middle Graph of Wheel Graph M(W1,n). Also we obtained the Wheel Graph W1,n and its Middle Graph M(W1,n) for n ≥ 3 are IBEDE Graceful Labeling . And the Wheel Graph W1,n for n > 3and its Middle Graph M(W1,n) for n ≥ 3 are SIBEDE Graceful Labeling. Here the Ramsey number of IBEDE and SIBEDE Graceful Labeling for W1,5 is 3 and the Ramsey number of IBEDE and SIBEDE Graceful Labeling for M(W1,3) is 18.

An L(2, 1)-labelling of a graph is a function f from the vertex set to the positive integers such that |f(x) - f(y)| ≥ 2 if dist(x, y) = 1 and |f(x) - f(y)| ≥ 1 if dist(x, y) = 2, where dist(x, y) is the distance between the two vertices x and y in the graph G. The span of an L(2, 1)-labelling f is the difference between the largest and the smallest labels used by f plus 1. In 1992, Griggs and Yeh conjectured that every graph with maximum degree Δ ≥ 2 has an L(2, 1)-labelling with span at most Δ2 + 1. By settling this conjecture for Δ sufficiently large, we prove the existence of a constant C such that the span of any graph of maximum degree Δ is at most Δ 2 + C.

Graph labeling is the concept of assigning identifiers to the nodes in a graph. The concept, being introduced in the late 1960s, has flourished much over time, and numerous papers have been written based on multiple labeling concepts. Graceful labeling, a particularly unique concept of graph labeling, was introduced by Rosa in 1967, which follows the idea that every vertex of a graph with $e$ edges can be assigned a unique label from the set { 0, 1, 2, …, $e$ }, and the labels of the edges will be assigned with the absolute difference between the labels of its adjoining edges, where every edge will have a unique label. Much research has been conducted to prove more and more classes are graceful, which itself has significant application in numerous fields, like road mapping, resource distribution, radio astronomy, cryptography, traffic routing, X-ray crystallography. In this paper, we introduce a class of graph, created from a collection of path graph and C4 cycle graph, terming it as P - n Legged C 4 - p’ graphs, and provide enough reference to prove that the derived class is graceful. We also provide mathematical expressions for every vertex in the graph, whilst considering multiple cases of path and chain length.

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.

A graceful labeling of a graph G with ‘q’ edges and vertex set V is an injection f: VG → {0,1,2,….q} with the property that the resulting edge labels are also distinct, where an edge incident with vertices u and v is assigned the label |fu – fv| . A graph which admits a graceful labeling is called a graceful graph. A Shell graph is defined as a cycle Cnwith n -3 chords sharing a common end point called the apex . Shell graphs are denoted as Cn, n- 3. A multiple shell is defined to be a collection of edge disjoint shells that have their apex in common. Hence a double shell consists of two edge disjoint shells with a common apex.

Graph labeling is the process of assigning labels to vertices and edges under certain conditions. This paper investigates the graceful local antimagic labeling of various graph families, excluding symmetric labelings, using computational experiments and Python-based algorithms. Through these experiments, we identify new results and patterns within specific graph classes. The study expands on the existing literature by offering computational evidence, proposing algorithms for the verification of labelings, and exploring the relationship between the local antimagic labeling and the chromatic number. Our results increase the understanding of graph labeling and offer insights into its computational aspects.

In this paper we show that a tree T with the following properties have graceful labeling. 1) T has a path H such that every pendant vertex of T has distance n (a fixed positive integer) from H . 2) Every vertex of T excluding one end vertex of H has even degree. Key words : Graceful labeling; n distant tree; Component moving transformation; Transfer of the first type

For fixed integers r,ell geq 0, a graph G is called an (r,ell)-graph if the vertex set V(G) can be partitioned into r independent sets and ell cliques. This brings us to the following natural parameterized questions: Vertex (r,ell)-Partization and Edge (r,ell)-Partization. An input to these problems consist of a graph G and a positive integer k and the objective is to decide whether there exists a set S subseteq V(G) (S subseteq E(G)) such that the deletion of S from G results in an (r,ell)-graph. These problems generalize well studied problems such as Odd Cycle Transversal, Edge Odd Cycle Transversal, Split Vertex Deletion and Split Edge Deletion. We do not hope to get parameterized algorithms for either Vertex (r, ell)-Partization or Edge (r,ell)-Partization when either of r or ell is at least 3 as the recognition problem itself is NP-complete. This leaves the case of r,ell in {1,2}. We almost complete the parameterized complexity dichotomy for these problems by obtaining the following results: - We show that Vertex (r,ell)-Partization is fixed parameter tractable (FPT) for r,ell in {1,2}. Then we design an Oh(sqrt log n)-factor approximation algorithms for these problems. These approximation algorithms are then utilized to design polynomial sized randomized Turing kernels for these problems. - Edge (r,ell)-Partization is FPT when (r,ell)in{(1,2),(2,1)}. However, the parameterized complexity of Edge (2,2)-Partization remains open. For our approximation algorithms and thus for Turing kernels we use an interesting finite forbidden induced graph characterization, for a class of graphs known as (r,ell)-split graphs, properly containing the class of (r,ell)-graphs. This approach to obtain approximation algorithms could be of an independent interest.

For a graph G=V,E, a subset F of E is called an edge dominating set of G if every edge not in F is adjacent to some edge in F. The edge domination number γ′G of G is the minimum cardinality taken over all edge dominating sets of G. Here, we determine the edge domination number for shadow graphs, middle graphs, and total graphs of paths and cycles.

Klein cordial trees and odd cyclic cordial friendship graphs

We introduce the notion of total distance irregular labeling, called the local total distance irregular labeling. All edges and vertices are labeled with positive integers 1 to k such that the weight calculated at the vertices induces a vertex coloring if two adjacent vertices has different weight. The weight of a vertex $u\in V(G)$ is defined as the sum of the labels of all vertices adjacent and edges incident to $u$ (distance $1$ from $u$). The minimum cardinality of the largest label over all such irregular assignment is called the local total distance irregularity strength, denoted by $tdis_l(G)$. In this paper, we established the lower bound of the local total distance irregularity strength of graphs $G$ and determined exact values of some classes of graphs namely path, cycle, star, bipartite complete, fan and sun graph.

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 ) .

Any graph that depicts a particular molecular structure can be given a topological graph index, also known as a molecular descriptor. From this index, it is possible to examine numerical data and learn more about some of the physical characteristics of a molecule. In this study, we construct and explore a novel variation for graphs termed the exponential Wiener index (EW(G)) and the multiplicative exponential Wiener index [Formula: see text]. Typical graphs for which it is also conceivable include the wheel graph, fan graph, star graph, full graph, cycle graph, complete graph, and complete bipartite graph. Additionally, some EW(G) bounds are determined using exponential inequalities. In the end, an algorithm is created to obtain EW(G), and the most effective is investigated. This work is also extended to the molecular graphs for n-alkanes, n = 1,2,..., 10 and benzenoid hydrocarbons. Additionally, a QSPR study that takes into account the molecular mass, melting point, and boiling point of alkanes and total π-electronic energy [Formula: see text], enthalpy of formation [Formula: see text], and boiling points have been conducted. The influential measures such as DFBETAS, DFFITS, Hadi's measure, are calculated by which the influential parameters and the outliers are identified for the regression models between the topological index and these physical parameters.

Let G = (V,E) be a graph with vertex set V and edge set E. The graph G is said to be a local irregular vertex coloring if there is a function f is a called a local irregularity vertex coloring if : (i) l : (V (G)) → {1, 2,…,k} as a vertex irregular k-labeling and w : V(G) → N, for every uv ∈ E(G),w(u) = w(v) where w(u) = Σv∈N(u) l(i) and (ii) opt (I) = min{rnax{li; livertex irregular labeling}}. The chromatic number of local irregularity vertex coloring of G denoted by Xlis(G), is the minimum cardinality of the largest label over all such local irregularity vertex coloring. In this paper, we study local irregular vertex coloring of G ⊳∘ Sn when G is complete bipartite graph (Km,p), ladder graph (Lm), fan graph (Fm), friendship graph (Frm), and cycle graph (Cm).