RETRACTED: On Cubic Roots Cordial Labeling for Some Graphs

Let be a graph with p vertices and q edges. We have introduced a new graph labeling method using integers and cordial-related works and investigated some graphs for this labeling technique. Using this labeling concept, we have examined the graphs like path, cycle, star, complete graph, comb, and wheel graph. The first research paper on graph theory was published by Leonhard Euler. However, he did not use the word ‘graph’ in his work. In the early stages of the development of the subject, the vertices of a graph were specified as , and the edges were denoted by, . In recent times, several researchers have attempted to provide different types of labeling to the vertices and edges of a graph by identifying the relevant mathematical properties. The present paper provides a novel method of labeling by employing integers, which may form a foundation for future research work. In this paper, we investigate the pair mean cordial labeling behavior of some cycle-related graphs like the ice cream graph, closed web graph, circulant graph, zig-zag chord graph, pentagonal circular ladder, djembe graph, quadrilateral friendship graph, and origami graph.

Cordial labeling of hypertrees

A k-edge product cordial labelling is a variant of the well-known cordial labelling. In this paper, a balanced k-edge product cordial labelling is suggested and some sufficient conditions for balanced 3-edge product cordial graphs are proved. Moreover, a construction of graphs admitting a balanced 3-edge product cordial labelling is presented.

The use of graph‐theoretic parameters is essential in interpreting the structural robustness, efficiency, and functionality of complex networks. Specifically, edge‐based metrics are more useful in interpreting the susceptibility of links and the concept of upstream impact, as well as structural robustness. Such metrics are essential in a variety of fields such as network security, communication, and optimization theory. In the same way, a new edge‐based invariant is introduced in this research paper, which is known as the b -edge differential . The b‐edge differential is essentially linked to b ‐differential of a graph. Let G = ( V , E ) be a simple connected graph. For any subset of edges X ⊆ E , we associate the set B ( X ) = { e ∈ E \ X : e shares a common vertex with at least one edge in X }, that is, B ( X ) captures all edges outside X which are adjacent to X . The quantity ∂ B E ( X ) = | B ( X )| is called the b ‐edge differential of X . We then define the b ‐edge differential of a graph G by ∂ B E ( G ) = max X ⊆ E ∂ B E ( X ). This parameter calculates the maximum value of the indirect dominances, or coverage, which can be achieved by a given set of edges. It serves as a useful tool in the identification of critical edges as well as the maximization of influence in networks. The problem can also be used in the applications involving the maximization of profit, traffic routing, and vulnerability evaluation for communication networks. We carry out the explicit computation of the b ‐edge differential for some classical types of graphs, such as paths, cyclic graphs, wheel graphs, complete graphs, star graphs, double‐star graphs, comb graphs, complete bipartite graphs, ladder graphs, and triangular ladder graphs. The work described in this article expands the relevance of differential parameters in the theory of graphs, with promising research directions for theoretical and application aspects of network science.

This paper presents first an analysis of the topology of mechanisms via Graph Theory and Screw Theory and next the principle of dual mechanisms in terms of their mobility and overconstraints. Using dual graphs, the graph representations of the mechanisms that are dual to hybrid and Delta mechanisms are revealed. The concept of parallel mechanisms with configurable platforms (PMCPs) is introduced and it is shown that the graph reduction of PMCPs always results in a wheel graph, which has the interesting property of being self-dual. In case of self dual topology, it is then possible to directly convert any method developed for their mobilty analysis into an overconstraint analysis method and vice versa. This self dual topology property can also be exploited to create new PMCPs and is an important aspect in the future development of a type synthesis method that includes PMCPs.KeywordsParallel mechanismsConfigurable platformTopologyMobilityOverconstraintsDuality

ABSTRACTAnalyzing complex multivalued datasets, such as boiling points and molecular Platt numbers, poses significant challenges due to the intricate relationships within the data, which traditional methods struggle to capture effectively. To address this, the study introduces Edge Neighborhood Graphs Ne(G), a novel graph‐based approach where the neighborhoods of edges in a graph G = (V, E) are treated as vertices, and an edge is drawn between two vertices if their corresponding neighborhoods intersect. This method is explored using complete, cycle, and wheel graphs, with formulas derived for vertex degrees in Ne(G) to uncover structural patterns. Polynomial approximations are employed to apply Ne(G) to multivalued data, demonstrating its effectiveness in modeling complex relationships and providing accurate predictions. By offering a powerful new tool for analyzing intricate datasets, this study makes a unique contribution to graph theory and data modeling, with promising potential for application to other graph types and diverse datasets in the future.

This dissertation solves two problems relating to the structure of graphs. The first of these is motivated by Kuratowski's Theorem, perhaps the most famous result in graph theory. This theorem states the $K\sb5$ and K$\sb{3,3}$ are the only non-planar graphs for which both $G\\ e$, the deletion of the edge e, and G/e, the contraction of the edge e, are planar for all edges e of G. We characterize the non-planar graphs for which $G\\ e$ or G/e is planar for all edges e of G. The second problem we solve is motivated by Tutte's wheels theorem. An immediate consequence of this theorem is that the wheel graphs are the basic building blocks for the collection of simple, 3-connected graphs. Therefore it is of interest to examine the structure of the graphs that do not have a minor isomorphic to the k-spoked wheel, $W\sb{k}$, for small values of k. Dirac determined that the graphs having no $W\sb3$-minor are the series-parallel networks. It follows easily from Tutte's wheels theorem is that $W\sb3$ is the only graph that has a $W\sb3$-minor and no $W\sb4$-minor. Oxley characterized the graphs that have a $W\sb4$-minor and no $W\sb5$-minor. We characterize the planar graphs that have a $W\sb5$-minor and no $W\sb6$-minor. We also determine the best-possible upper bound on the number of edges of such a graph.

Two $2$-cell embeddings $\imath : X \to S$ and $\jmath : X \to S$ of a connected graph $X$ into a closed orientable surface $S$ are congruent if there are an orientation-preserving surface homeomorphism $h : S \to S$ and a graph automorphism $\gamma$ of $X$ such that $\imath h =\gamma\jmath$. Mull et al. [Proc. Amer. Math. Soc. 103(1988) 321–330] developed an approach for enumerating the congruence classes of $2$-cell embeddings of a simple graph (without loops and multiple edges) into closed orientable surfaces and as an application, two formulae of such enumeration were given for complete graphs and wheel graphs. The approach was further developed by Mull [J. Graph Theory 30(1999) 77–90] to obtain a formula for enumerating the congruence classes of $2$-cell embeddings of complete bipartite graphs into closed orientable surfaces. By considering automorphisms of a graph as permutations on its dart set, in this paper Mull et al.'s approach is generalized to any graph with loops or multiple edges, and by using this method we enumerate the congruence classes of $2$-cell embeddings of a bouquet of circles and a dipole into closed orientable surfaces.

In this work, we introduce labeling techniques for the double star graph, a basic graph theory structure. In particular, we concentrate on calculating two different kinds of labeling: fortunate labeling and correct labeling. Adhering to the restrictions of vertex labeling in graph theory, appropriate labeling guarantees that neighboring vertices receive unique labels. A more recent and interesting idea is the fortunate labelling, which gives vertices positive numbers so that the labels on neighboring vertices add up to a unique value for each edge. We calculate and examine various labeling methods using algorithmic methods, proving their usefulness and effectiveness for the double star graph. Our findings further enhance the theoretical and practical capabilities of various labeling systems by offering insightful information about how to optimize them for bipartite and tree-like graphs.

Dominant local metric dimension is consist of two interesting topic in graph theory, they are dominating and metric dimension which be expanded as local metric dimension. A graph G is said having dominant local metric dimension if G be a connected graph and there is an ordered subset W l = {W 1, W 2 …, W n }⊆V(G) where W l is a local resolving set as well as a dominating set of G. Minimum cardinality of dominant local resolving set of G is called the dominant local basis of G. Then, cardinality of the dominant local basis of G is called the dominant local metric dimension of G which denoted by Ddim l (G). In this paper, we determine the dominant local metric dimension of wheel related graphs. That are Wheel, Jahangir, Friendship and Helm graph. Furthermore, we characterize the dominant local metric dimension of those graphs.

In this paper we prove that the split graphs of K1,n and Bn,n are prime cordial graphs. We also show that the square graph of Bn,n is a prime cordial graph while middle graph of Pn is a prime cordial graph for n≥4 . Further we prove that the wheel graph Wn admits prime cordial labeling for n≥8.

In this paper, a technique of coding a message using GMJ code on the Fibonacci web FW(t, n) by applying the difference cordial labeling is provided. A few observations and a rule for labeling on the Fibonacci Web are given. Two illustrations using one method for two messages are dealt with in detail in this paper. The catchy clue, mathematical or non mathematical, using numbers and words for finding the suitable Fibonacci Web and GMJ coding given are meant to maintain secrecy and to induce interest. Pictorial coding is also done.

The study of valuations of graphs is a relatively young part of graph theory. In this article we survey what is known about certain graph valuations, that is, labeling methods: antimagic labelings, edge-magic total labelings and vertex-magic total labelings.

Soft graph theory offers a parametrized perspective on graphs by classifying the universe’s components according to a specified set of parameters. This paper explores soft graph theory, which classifies graph components based on a set of parameters, focusing on the wheel and friendship graph families. By analyzing soft graphs with distance-based parameter sets, the study provides key results on the isomorphic subgraphs formed within these structures. These findings offer important insights into the structure and behavior of soft graphs, enhancing our understanding of soft graph theory.

Throughout this study we have shown certain graphs such as Z − Pn, subdivision of Z − Pn, braid graph B(n), duplication of leaf node (pendant vertex) by an vertex and by an edge of Z−Pn are difference cordial graphs.