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This study focuses on finding super edge-magic total (SEMT) labeling and deficiency of imbalanced fork and disjoint union of imbalanced fork with star, bistar and path; in addition, the SEMT strength for Imbalanced Fork is investigated.

A graph ℘ is said to be edge-magic total (EMT if there is a bijection Υ : V(℘) ∪ E(℘) → {1, 2, …, |V(℘) ∪ E(℘)|} s.t., Υ(υ) + Υ(υν) + Υ(ν) is a constant for every edge υν ∈ E(℘). An EMT graph ℘ will be called strong edge-magic total (SEMT) if Υ(V(℘)) = {1, 2, …, |V(℘)|}. The SEMT strength, sm(℘), of a graph ℘ is the minimum of all magic constants a(Υ), where the minimum runs over all the SEMT valuations of ℘, this minimum is defined only if the graph has at least one such SEMT valuation. Furthermore, the SEMT deficiency of a graph ℘, μs(℘), is either the minimum non-negative integer n such that ℘ ∪ nK1 is SEMT or +∞ if there will be no such integer n. In this paper, we will present the strong edge-magicness and deficiency of disjoint union of 2-sided generalized comb with bistar, path and caterpillar, moreover we will evaluate the SEMT strength for 2-sided generalized comb.

A super edge-magic total (SEMT) labeling of a graph ℘(V, E) is a one-one map ϒ from V(℘)∪E(℘) onto {1, 2,…,|V (℘)∪E(℘) |} such that ∃ a constant “a” satisfying ϒ(υ) + ϒ(υν) + ϒ(ν) = a, for each edge υν ∈E(℘), moreover all vertices must receive the smallest labels. The super edge-magic total (SEMT) strength, sm(℘), of a graph ℘ is the minimum of all magic constants a(ϒ), where the minimum runs over all the SEMT labelings of ℘. This minimum is defined only if the graph has at least one such SEMT labeling. Furthermore, the super edge-magic total (SEMT) deficiency for a graph ℘, signified as μs(℘)$\mu_{s}(\wp)$ is the least non-negative integer n so that ℘∪nK1 has a SEMT labeling or +∞ if such n does not exist. In this paper, we will formulate the results on SEMT labeling and deficiency of fork, H -tree and disjoint union of fork with star, bistar and path. Moreover, we will evaluate the SEMT strength for trees.

An edge-magic total labeling of an (n,m)-graph G = (V,E) is a one to one map λ from V(G) ∪ E(G) onto the integers {1,2,…,n + m} with the property that there exists an integer constant c such that λ(x) + λ(y) + λ(xy) = c for any xy ∈ E(G). It is called super edge-magic total labeling if λ (V(G)) = {1,2,…,n}. Furthermore, if G has no super edge-magic total labeling, then the minimum number of vertices added to G to have a super edge-magic total labeling, called super edge-magic deficiency of a graph G, is denoted by μs(G) [4]. If such vertices do not exist, then deficiency of G will be + ∞. In this paper we study the super edge-magic total labeling and deficiency of forests comprising of combs, 2-sided generalized combs and bistar. The evidence provided by these facts supports the conjecture proposed by Figueroa-Centeno, Ichishima and Muntaner-Bartle [2].

It has been observed that there is a strong correlation between topological descriptors of chemical structures and their physical properties in a number of tests every year. There are several of these characteristics, including molecular mass, density, melting and boiling points, flash points, and others. Topological indices are numerical numbers associated with the molecular graph and are used in quantitative structure property relationship studies to predict their physical and chemical properties. In this work, we have calculated the reverse degree based topological descriptors of five molecular structures namely, hyaluronic acid–paclitaxel conjugates (Gn), anticancer drug (SPn), polyomino chain of n‐cycle (Zn), triangular benzenoid (Tn), and circumcoronene benzenoid series (Hk,k≥1). The values of topological indices are used in entropy method to calculate the weights. Then, using multiple criteria decision‐making techniques like COPRAS and PROMETHEE, we rank these molecular structures. [ABSTRACT FROM AUTHOR]

Topological indices provide a mathematical language for capturing molecular structure, symmetry, and predicting properties. Dendrimers are microscopic bilaterally symmetrical molecules with a well-defined homogeneous nanoparticles structure, often consisting of a symmetric center, inner shell, and outer shell. In this work, first we compute some degree-based topological indices of Porphyrin (DnPn),Poly (Propyl) Ether Imine(PETIM), Zinc porphyrin (DPZn), and Polyamidoamine (PAMAM) dendrimers. Then, we use multi-criteria decision making (MCDM) techniques to establish the weighted evaluation of dendrimer classes based on certain topological indices. For weighted analysis we correlate the properties of benzene derivatives with topological invariants. Finally, based on the multi-criteria decision making techniques namely TOPSIS, SAW and MOORA method, we have ranked the dendrimer structures based on their properties. [ABSTRACT FROM AUTHOR]

Phenylenes belong to a special class of conjugated hydrocarbons composed of a special arrangement of hexagons and squares. In the phenylene structure, every square is adjacent to a pair of hexagons, and any two hexagons are not adjacent. If in the structure of phenylene only two squares are adjacent to every hexagon, then it is called a phenylene chain. In this paper, we compute the expected values of the first and second Gourava index, the first and second redefined Zagreb index, and the Hyper-Zagreb index of random phenylene chains. Furthermore, a comparison between the expected values of the computed indices among the classes of random phenylene chains has been investigated. [ABSTRACT FROM AUTHOR]

Graph theory has found significant use in chemistry, especially in modelling chemical structures. A molecular graph can be represented by a graph where atoms represent vertices and links between these atoms are called edges. Each molecular structure has its own physical properties, such as boiling point, melting point, density, molecular mass, entropy, etc. A topological invariant of a molecular structure is a descriptor that is correlated to certain chemical and biological actions of the chemical compounds. In this paper, we studied the structure of oxide networks, triangular silicate networks, dominating oxide networks and dominating silicate networks in connection to some additive degree-based topological indices. Specifically, the ranking among these networks is also determined with the help of certain multi-criteria decision-making techniques (MCDM). In the end, regression and correlation analyses between their chemical properties and topological indices has been observed. [ABSTRACT FROM AUTHOR]

The classification of contact simple map germs from (C 2 , 0) → (C 2 , 0) was given by Dimca and Gibson. In this article, we give a useful criteria to recognize this classification of contact simple map germs of holomorphic mappings with finite codimension. The recognition is based on the computation of explicit numerical invariants. By using this characterization, we implement an algorithm to compute the type of the contact simple map germs without computing the normal form and also give its implementation in the computer algebra system Singular. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the degree based topological properties of the novel planar metal-organic networks TM – TCNB. Interestingly, the TM – TCNB systems are metallic in any event in one turn heading and show long-run ferromagnetic coupling on the off chance that for attractive structures, which speak to perfect applicants and a fascinating possibility of uncommon applications in spintronics. Chemical graph theory plays an important role in modeling and designing any chemical structure. The topological indices are the numerical invariants of a molecular graph and are very useful for predicting their physical properties. We have computed the degree based topological indices for this metal-organic networks TM – TCNB. [ABSTRACT FROM AUTHOR]

The F -index of a graph Q is defined as F Q = ∑ t ∈ V Q d t 3 . In this paper, we use edge swapping transformations to find the extremal value of the F-index among the class of trees with given order, pendent vertices, and diameter. We determine the trees with given order, pendent vertices, and diameter having the greatest F -index value. Also, the first five maximum values of F index among the class of trees with given diameter are determined. [ABSTRACT FROM AUTHOR]

Let G be a simple connected graph. The second Gourava index of graph G is defined as G O 2 G = ∑ θ ϑ ∈ E G d θ + d ϑ d θ d ϑ where d θ denotes the degree of vertex θ. If removal of a vertex of G forms a tree, then G is called an apex tree. Let L ⊂ V G with ∣ L ∣ = q. If removal of L from V G forms a tree and any other subset of V G whose cardinality is less than ∣ L ∣ does not form a tree, then G is known as q -apex tree. In this paper, we have calculated upper bound for 2 nd Gourava index with respect to q -apex trees. [ABSTRACT FROM AUTHOR]

A set of nodes called vertices V accompanied with the lines that bridge these nodes called edges E compose an explicit figure termed as a graph G(V, E). |V (G)| = v and |E(G)| = ε specify its order and size respectively. A (v, ε)-graph G determines an edge-magic total (EMT) labeling when ... is bijective so as the weights at every edge are the same constant (say) c i.e., for x, y ∈ V (G); Γ(x) + Γ(xy) + Γ(y) = c, independent of the choice of any xy ∈ E(G), such a number is interpreted as a magic constant. If all vertices gain the smallest of the labels then an EMT labeling is called a super edge-magic total (SEMT) labeling. If a graph G allows at least one SEMT labeling then the smallest of the magic constants for all possible distinct SEMT la-belings of G describes super edge-magic total (SEMT) strength, sm(G), of G. For any graph G, SEMT deficiency is the least number of isolated vertices which when uniting with G yields a SEMT graph. In this paper, we will find SEMT labeling and deficiency of forests consisting of two components, where one of the components for each forest is generalized comb .. And other component is a star, bistar, comb or path respectively, moreover, we will investigate SEMT strength of aforesaid generalized comb. [ABSTRACT FROM AUTHOR]

A Graph G = (V(G), E(G)) is the set of points called vertices or(nodes) and the lines connecting these points are called edges. The number of vertices in a graph is called its order and the number of edges is called its size, usually denoted as |V(G)| = n (orp) and |E(G)| = m (or q) respectively. A graph G with p nodes and q lines admits the edge magic total labeling if there exists a one-one, onto map Ψ : V (G) U E (G) → {1,p + q} = {1,2,3,... ,p + q}. s.t weight of every edge is some same constant (say)k, such number k is called the magic constant. If a graph G has an edge magic total labeling Ψ : V(G) → {1,2,3,... ,p} then Ψ is called super edge magic total(SEMT) labeling. For graph G, SEMD is the number of isolated vertices whose union with G makes the resulting graph SEMT. μ8(G), is the minimum non-negative integer n such that G U nK1 SEMD will be +∞ if no isolated vertex do this job. In this work SEMT labeling and deficiencies are determined for forests formed by two sided generalized combs, stars, combs and banana trees. [ABSTRACT FROM AUTHOR]

The F -index of a graph Q is defined as F ( Q )=∑ t ∈ V ( Q ) ( d t ) 3 . In this paper, we use edge swapping transformations to find the extremal value of the F-index among the class of trees with given order, pendent vertices, and diameter. We determine the trees with given order, pendent vertices, and diameter having the greatest F -index value. Also, the first five maximum values of F index among the class of trees with given diameter are determined., Competing Interests: The authors declare that they have no conflicts of interest., (Copyright © 2022 Wenhu Wang et al.)