Graphical Invariants Research Articles

Graphical Invariants for some Transformed Networks

A topological index is a numerical character associated with a graph that is invariant under graph isomorphism and describe the graphs topology. There are several graph operations that may be used to change it into a new structure, such as constructing a stellation, bounded dual, complement, subdivided, line graph, minor, dual, and medial. In this paper, we construct transformed networks from the concealled non-kekulean benzenoid hydrocarbon structure by applying stellation and bounded dual operations, further we study their degree based topological properties by appropriately labeling the graph. Degree based topological indices are playing significant role among other types of indices in chemical, pharmaceutical and bio-informatics industry, since they correlate the structure with its physicochemical properties.

On Molecular Descriptors of Polycyclic Aromatic Hydrocarbon

A variety of graphical invariants have been described and tested, offering lots of applications in the fields of nanochemistry, computational networks, and different scientific research areas. One commonly studied group of invariants is the topological index, which allows research on the chemical, biological, and physical properties of a chemical structure. Topological indices are numerical quantities that can be used to describe the properties of the molecular graph. In this article, we draw from the analytically closed formulas of molecular structures of polycyclic aromatic hydrocarbons by calculating temperature-based topological indices. Our results in this paper may be useful to better understand many physical and chemical properties of polycyclic aromatic hydrocarbons.

Topological Properties of some Benzenoid Chemical Structures by Using Face Index

A variety of graphical invariants have been described and tested, offering lots of applications in the fields of Nano chemistry, computational networks and different scientific research areas. One commonly studied group of invariants is the topological index, which allows the prediction of the chemical, biological and physical properties of a chemical structure. Topological indexes are numerical quantities that can be used to describe the properties of the molecular graph. In this article, we computed the face index formula for certain molecular structures of benzenoid series such as polycyclic aromatic hydrocarbon (PAH), jagged-rectangle benzenoid system (JRB[m,n]) and the series of concealed non-kekulean benzenoid system.

Ve-degree and Ev-degree Based Topological Properties of Magnesium Oxide MgO (111) Structures

The topological index is a numerical representation of a molecular structure and predicts the physical property of a chemical compound. In the present work, we have developed explicit expressions of recently defined novel Ve- degrees and Ev-degrees based topological indices of Magnesium Oxide MgO (111). For this by consider a unit cell of MgO (111) which is extended to its general structure. After that, we have computed Ve-degrees and Ev-degrees of this general structure of MgO (111) then by employing well-known topological indices on these computed Ve- degrees and Ev-degrees of the general structure, obtained numerical results. A variety of graphical invariants have been described and tested, offering lots of applications in the fields of nano-chemistry, computational networks, and indifferent scientific research areas. These computed results will help study the physical, chemical, and biological properties of the structure such as boiling, melting, and flickering point; moisture and forming heat; temperature, strain, density, etc.

On Analysis of Temperature Based Topological Indices of Some Covid-19 Drugs

A variety of graphical invariants have been described and tested, offering lots of applications in the fields of nanochemistry, computational networks and in different scientific research areas. One commonly studied group of invariants is the topological index, which allows to research the chemical, biological, and physical properties of a chemical structure. Topological indexes are numerical quantities that can be used to describe the properties of the molecular graph. In this article, we draw from the analytically closed formulas of certain molecular structures of coronavirus such as Ribavirin, Sofosbuvir and Oseltamivir by calculating temperature based topological indices.

Generalized Second Neighborhood Zagreb Index: Mathematical Inequalities and Chemical Applicability of PAHs

Topological indices are graphical invariants that relate a numeric number to a graph, which is structurally invariant and predicts the chemical, biological and physical features of the molecular graphs. In this work, mathematical inequalities of generalized second neighborhood Zagreb index are obtained. Further, generalized second neighborhood Zagreb indices for some particular values are computed for some basic polycyclic aromatic hydrocarbons, and the QSPR analysis are also obtained.

On the depth of binomial edge ideals of graphs

Let G be a graph on the vertex set [n] and $$J_G$$ the associated binomial edge ideal in the polynomial ring $$S=\mathbb {K}[x_1,\ldots ,x_n,y_1,\ldots ,y_n]$$ . In this paper, we investigate the depth of binomial edge ideals. More precisely, we first establish a combinatorial lower bound for the depth of $$S/J_G$$ based on some graphical invariants of G. Next, we combinatorially characterize all binomial edge ideals $$J_G$$ with $$\mathrm {depth} S/J_G=5$$ . To achieve this goal, we associate a new poset $$\mathscr {M}_G$$ with the binomial edge ideal of G and then elaborate some topological properties of certain subposets of $$\mathscr {M}_G$$ in order to compute some local cohomology modules of $$S/J_G$$ .

Comparative Study of Y‐Junction Nanotubes with Vertex‐Edge Based Topological Descriptors

The current results of various forms of carbon nanostructures and its applications in different areas attract the researchers. In pharmaceutical, medicine, industry and electronic devices they used it by its graphical invariants. The detection of different types of carbon nanotubes junctions enhanced the attention and interest for forthcoming devices like transistors and amplifiers. A topological index plays a very important role in the study of physicochemical properties of biological and chemical structures. In this paper, we determine results of ve‐degree topological indices for various type of carbon nanotubes Y‐junctions and their comparisons. The particular indices called as The first ve‐degree Zagreb β index, the second ve‐degree Zagreb index, ve‐degree Randic index, ve‐degree atom‐bond connectivity index, ve‐degree geometric‐arithmetic index, ve‐degree harmonic index and ve‐degree sum‐connectivity index.

On the first Zagreb index and multiplicative Zagreb coindices of graphs

Abstract For a (molecular) graph G with vertex set V (G) and edge set E(G), the first Zagreb index of G is defined as , where dG(vi) is the degree of vertex vi in G. Recently Xu et al. introduced two graphical invariants and named as first multiplicative Zagreb coindex and second multiplicative Zagreb coindex, respectively. The Narumi-Katayama index of a graph G, denoted by NK(G), is equal to the product of the degrees of the vertices of G, that is, NK(G) = . The irregularity index t(G) of G is defined as the number of distinct terms in the degree sequence of G. In this paper, we give some lower and upper bounds on the first Zagreb index M1(G) of graphs and trees in terms of number of vertices, irregularity index, maxi- mum degree, and characterize the extremal graphs. Moreover, we obtain some lower and upper bounds on the (first and second) multiplicative Zagreb coindices of graphs and characterize the extremal graphs. Finally, we present some relations between first Zagreb index and Narumi-Katayama index, and (first and second) multiplicative Zagreb index and coindices of graphs.

On the multiplicative Zagreb coindex of graphs

For a (molecular) graph \(G\) with vertex set \(V(G)\) and edge set \(E(G)\), the first and second Zagreb indices of \(G\) are defined as \(M_1(G) = \sum_{v \in V(G)} d_G(v)^2\) and \(M_2(G) = \sum_{uv \in E(G)} d_G(u)d_G(v)\), respectively, where \(d_G(v)\) is the degree of vertex \(v\) in \(G\). The alternative expression of \(M_1(G)\) is \(\sum_{uv \in E(G)}(d_G(u) + d_G(v))\). Recently Ashrafi, Doslic and Hamzeh introduced two related graphical invariants \(\overline{M}_1(G) = \sum_{uv \notin E(G)}(d_G(u)+d_G(v))\) and \(\overline{M}_2(G) = \sum_{uv \notin E(G)} d_G(u)d_G(v)\) named as first Zagreb coindex and second Zagreb coindex, respectively. Here we define two new graphical invariants \(\overline{\Pi}_1(G) = \prod_{uv \notin E(G)}(d_G(u)+d_G(v))\) and \(\overline{\Pi}_2(G) = \prod_{uv \notin E(G)} d_G(u)d_G(v)\) as the respective multiplicative versions of \(\overline{M}_i\) for \(i = 1, 2\). In this paper, we have reported some properties, especially upper and lower bounds, for these two graph invariants of connected (molecular) graphs. Moreover, some corresponding extremal graphs have been characterized with respect to these two indices.

QSAR of anti tuberculosis drugs of INH type using graphical invariants

Quantitative structure–activity relationship studies have been performed on a series of analogues of tuberculostatic drugs of isonicotinic acid hydrazide type using multi parameter regression analysis. Theoretical molecular descriptors like Wiener index, molecular complexity indices and inter molecular similarity have been introduced in the physicochemical property based models and the superiority of the models over property–property correlation is clearly established. Theoretical molecular descriptors can be calculated directly from molecular structure, so its use in QSAR models is of immense utility compared to experimentally determined physicochemical parameters that are not always available for a particular structure. The influence of non-empirical parameters on the activity of INH and its derivatives are discussed in details. The justification for the choice of molecular descriptors based on ridge regression has been made and linearity of the models discussed.

On the use of graph invariants for efficiently generating hydrogen bond topologies and predicting physical properties of water clusters and ice

Water clusters and some phases of ice are characterized by many isomers with similar oxygen positions, but which differ in direction of hydrogen bonds. A relationship between physical properties, like energy or magnitude of the dipole moment, and hydrogen bond arrangements has long been conjectured. The topology of the hydrogen bond network can be summarized by oriented graphs. Since scalar physical properties like the energy are invariant to symmetry operations, graphical invariants are the proper features of the hydrogen bond network which can be used to discover the correlation with physical properties. We demonstrate how graph invariants are generated and illustrate some of their formal properties. It is shown that invariants can be used to change the enumeration of symmetry-distinct hydrogen bond topologies, nominally a task whose computational cost scales like N2, where N is the number of configurations, into an N ln N process. The utility of graph invariants is confirmed by considering two water clusters, the (H2O)6 cage and (H2O)20 dodecahedron, which, respectively, possess 27 and 30 026 symmetry-distinct hydrogen bond topologies associated with roughly the same oxygen atom arrangements. Physical properties of these clusters are successfully fit to a handful of graph invariants. Using a small number of isomers as a training set, the energy of other isomers of the (H2O)20 dodecahedron can even be estimated well enough to locate phase transitions. Some preliminary results for unit cells of ice-Ih are given to illustrate the application of our results to periodic systems.

Edge-disjoint minimum-weight connected spanning k-edge subgraphs in a weighted graph: A connectedness theorem

Farber et al. (1985) proved that any pair of edge-disjoint spanning trees in a graph can be obtained from any other by a sequence of single-edge exchanges in a way that preserves, at each step, the property of being edge-disjoint spanning trees. In this paper, we consider a generalization of this problem concerning pairs of edge-disjoint minimum-weight connected spanning k-edge subgraphs in a weighted graph. It is shown that any pair of edge-disjoint minimum-weight connected spanning k-edge subgraphs of a weighted graph can be obtained from any other by a sequence of single-edge exchanges in a way that preserves, at each step, the property of being edge-disjoint minimum-weight connected spanning k-edge subgraphs. As an application, we give a two-dimensional interpolating theorem for some graphical invariants.

The evaluation of spectral moments for molecular graphs of phenylenes

The evaluation of some spectral moments for phenylenes is considered. Explicit topological formulae for the fourth, sixth and eighth moments are derived. This was achieved by adopting the method of Hall [1], which was originally applied to benzenoid systems, and applying it to molecular graphs of phenylenes. It is shown that the moments considered in this paper can be expressed in terms of three mutually independent graphical invariants.

Sequences of graphical invariants

AbstractFor a given graphical invariant π, a sequence (ao, a1,…, an) of positive integers is said to be π‐feasible if there exists a graph G with distinguished vertices ν1, ν2,…, νn such that π(G) = ao and π(G −ν1 −ν2 −…−νi) = ai for i = 1, 2,…,n. In this paper, we investigate π‐feasible sequences for the irredundance, domination, and independence numbers of a graph.

The evaluation of the eighth moment for benzenoid graphs

The evaluation of the eighth moment of the adjacency matrix of benzenoid graphs is considered. It is found that the eighth moment can be expressed in terms of 7 graphical invariants. By this we extend the recently obtained results of Hall [1] and Dias [2].

INGRID: A graph invariant manipulator

INGRID, a software system for assisting researchers and students of graph theory, is described. It is a partial commutative rule-based system using forward chaining, consists of approximately 1200 production rules, and is written in Berkeley Pascal on a VAX 11/780 under UNIX. INGRID has a data base of 36 graphical invariants and approximately 350 theorems symbolically relating these invariants. A set of one or more user-supplied restrictions upon the values of some invariants is propagated by INGRID, via the theorems, into intervals of possible values for other of the invariants. With the simple user interface provided by INGRID, even a graph theory novice can often discern properties of a graph that might normally require the capabilities of a well-informed expert.

On the size of graphs of a given bandwidth

The relationship between the graphical invariants bandwidth and number of edges is considered. Bounds, some sharp and others improvements of known results, are given for the number of edges of graphs having a given bandwidth.

Molecules with holes

The bual and laub operations are defined and shown to be a useful means of analysing and coordinating previous descriptions of polyhexes. Some implications of a polyhex having a hole are explored. The position of such molecules in the Dias periodic table is argued. Formulae for moments of the Huckel energy in terms of graphical invariants are derived. The change in π-electron energy when a hole is formed is calculated for some molecules.

The evaluation of moments for polycyclic hydrocarbons

The evaluation of some moments of the energy in the Huckel theory of conjugated molecules is considered. It is shown that, for molecules consisting entirely of hexagons, the moments μ4 and μ6 can be expressed in terms of four graphical invariants. Partial results are given for other molecules. Since the total energy can be expressed as a series of moments the implications for the energy are discussed. In this discussion two other invariants play a major role. The conclusion is suggested that an analysis of moments in terms of graphical invariants should be of prime importance in understanding these molecules.