Search Results (25)

An algebraic graph is defined in terms of graph theory as a graph with related algebraic structures or characteristics. If the vertex set of a graph $\mathcal{G}$ is a group, a ring, or a field, then $\mathcal{G}$ is called an algebraic structure graph. This work uses an algebraic structure graph based on the modular ring ${\mathbb{Z}}_n$, known as a hyper-chordal ring network. The lower and upper bounds of the local fractional metric dimension are computed for certain families of hyper-chordal ring networks. Utilizing the cardinalities of local fractional resolving sets, local fractional resolving (LFR)M-polynomials are computed for hyper-chordal ring networks. Further, new topological indices based on (LFR)M-polynomials are established for the proposed networks. The local fraction entropies are developed by modifying the first three kinds of Zagreb entropies, which are calculated for the chosen hyper-chordal ring networks. Furthermore, numerical and graphical comparisons are discussed to observe the order between newly computed topological indices. Full article

The field of mathematics that studies the relationship between algebraic structures and graphs is known as algebraic graph theory. It incorporates concepts from graph theory, which examines the characteristics and topology of graphs, with those from abstract algebra, which deals with algebraic structures such as groups, rings, and fields. If the vertex set of a graph $\widehat{G}$ is fully made up of the zero divisors of the modular ring ${\mathbb{Z}}_n$, the graph is said to be a zero-divisor graph. If the products of two vertices are equal to zero under (modn), they are regarded as neighbors. Entropy, a notion taken from information theory and used in graph theory, measures the degree of uncertainty or unpredictability associated with a graph or its constituent elements. Entropy measurements may be used to calculate the structural complexity and information complexity of graphs. The first, second and second modified Zagrebs, general and inverse general Randics, third and fifth symmetric divisions, harmonic and inverse sum indices, and forgotten topological indices are a few topological indices that are examined in this article for particular families of zero-divisor graphs. A numerical and graphical comparison of computed topological indices over a proposed structure has been studied. Furthermore, different kinds of entropies, such as the first, second, and third redefined Zagreb, are also investigated for a number of families of zero-divisor graphs. Full article

A topological index, which is a number, is connected to a graph. It is often used in chemometrics, biomedicine, and bioinformatics to anticipate various physicochemical properties and biological activities of compounds. The purpose of this article is to encourage original research focused on topological graph indices for the drugs azacitidine, decitabine, and guadecitabine as well as an investigation of the genesis of symmetry in actual networks. Symmetry is a universal phenomenon that applies nature’s conservation rules to complicated systems. Although symmetry is a ubiquitous structural characteristic of complex networks, it has only been seldom examined in real-world networks. The $\overline{M}$-polynomial, one of these polynomials, is used to create a number of degree-based topological coindices. Patients with higher-risk myelodysplastic syndromes, chronic myelomonocytic leukemia, and acute myeloid leukemia who are not candidates for intense regimens, such as induction chemotherapy, are treated with these hypomethylating drugs. Examples of these drugs are decitabine (5-aza-20-deoxycytidine), guadecitabine, and azacitidine. The $\overline{M}$-polynomial is used in this study to construct a variety of coindices for the three brief medicines that are suggested. New cancer therapies could be developed using indice knowledge, specifically the first Zagreb index, second Zagreb index, F-index, reformulated Zagreb index, modified Zagreb, symmetric division index, inverse sum index, harmonic index, and augmented Zagreb index for the drugs azacitidine, decitabine, and guadecitabine. Full article

Topological indices are numbers that are applied to a graph and can be used to describe specific graph properties through algebraic structures. Algebraic graph theory is a helpful tool in a range of chemistry domains. Because it helps explain how the different symmetries of molecules and crystals affect their structure and dynamics, it is a powerful theoretical approach for forecasting both the common and uncommon characteristics of molecules. A topological index converts the chemical structure into a number and contributes a lot in chemical graph theory. In this article, we compute the Wiener index, Zagreb indexes, Wiener polynomial, Hyper-Wiener index, ABC index and eccentricity-based topological index of a nonzero component union graph from vector space. Full article

A topological index as a graph parameter was obtained mathematically from the graph’s topological structure. These indices are useful for measuring the various chemical characteristics of chemical compounds in the chemical graph theory. The number of atoms that surround an atom in the molecular structure of a chemical compound determines its valency. A significant number of valency-based molecular invariants have been proposed, which connect various physicochemical aspects of chemical compounds, such as vapour pressure, stability, elastic energy, and numerous others. Molecules are linked with numerical values in a molecular network, and topological indices are a term for these values. In theoretical chemistry, topological indices are frequently used to simulate the physicochemical characteristics of chemical molecules. Zagreb indices are commonly employed by mathematicians to determine the strain energy, melting point, boiling temperature, distortion, and stability of a chemical compound. The purpose of this study is to look at valency-based molecular invariants for $Si{O}_4$ embedded in a silicate chain under various conditions. To obtain the outcomes, the approach of atom–bond partitioning according to atom valences was applied by using the application of spectral graph theory, and we obtained different tables of atom—bond partitions of $Si{O}_4$. We obtained exact values of valency-based molecular invariants, notably the first Zagreb, the second Zagreb, the hyper-Zagreb, the modified Zagreb, the enhanced Zagreb, and the redefined Zagreb (first, second, and third). We also provide a graphical depiction of the results that explains the reliance of topological indices on the specified polynomial structure parameters. Full article

Gutman and Trinajstić (1972) defined the connection-number based Zagreb indices, where connection number is degree of a vertex at distance two, in order to find the electron energy of alternant hydrocarbons. These indices remain symmetric for the isomorphic (molecular) networks. For the prediction of physicochemical and symmetrical properties of octane isomers, these indices are restudied in 2018. In this paper, first and second Zagreb connection coindices are defined and obtained in the form of upper bounds for the resultant networks in the terms of different indices of their factor networks, where resultant networks are obtained from two networks by the product-related operations, such as cartesian, corona, and lexicographic. For the molecular networks linear polynomial chain, carbon nanotube, alkane, cycloalkane, fence, and closed fence, first and second Zagreb connection coindices are computed in the consequence of the obtained results. An analysis of Zagreb connection indices and coindices on the aforesaid molecular networks is also included with the help of their numerical values and graphical presentations that shows the symmetric behaviour of these indices and coindices with in certain intervals of order and size of the under study (molecular) networks. Full article

Given any function $f:{\mathbb{Z}}^{+}\to {\mathbb{R}}^{+}$ , let us define the f-index $I_f\left(G\right)={\sum }_{u\in V\left(G\right)}f\left(d_u\right)$ and the f-polynomial $P_f(G,x)={\sum }_{u\in V\left(G\right)}{x}^{1/f\left(d_u\right)-1},$ for $x>0$ . In addition, we define $P_f(G,0)={lim}_{x\to 0^{+}}{P}_f(G,x)$ . We use the f-polynomial of a large family of topological indices in order to study mathematical relations of the inverse degree, the generalized first Zagreb, and the sum lordeg indices, among others. In this paper, using this f-polynomial, we obtain several properties of these indices of some classical graph operations that include corona product and join, line, and Mycielskian, among others. Full article

Owing to their distinguished properties, titanium difluoride (TiF₂) and the crystallographic structure of Cu₂O have attracted a great deal of attention in the field of quantitative structure–property relationships (QSPRs) in recent years. A topological index of a diagram (G) is a numerical quantity identified with G which portrays the sub-atomic chart G. In 1972, Gutman and Trinajstić resented the first and second Zagreb topological files of atomic diagrams. In this paper, we determine a hyper-Zagreb list, a first multiple Zagreb file, a second different Zagreb record, and Zagreb polynomials for titanium difluoride (TiF₂) and the crystallographic structure of Cu₂O. Full article

A topological index of a graph is a single numeric quantity which relates the chemical structure with its underlying physical and chemical properties. Topological indices of a nanosheet can help us to understand the properties of the material better. This study deals with computation of degree-dependent topological indices like the Randic index, first Zagreb index, second Zagreb index, geometric arithmetic index, atom bond connectivity index, sum connectivity index and hyper Zagreb index of nanosheet covered by C₃ and C₆. Furthermore, M-polynomial of the nanosheet is also computed, which provides an alternate way to express the topological indices. Full article

Topological indices have been computed for various molecular structures over many years. These are numerical invariants associated with molecular structures and are helpful in featuring many properties. Among these molecular descriptors, the eccentricity connectivity index has a dynamic role due to its ability of estimating pharmaceutical properties. In this article, eccentric connectivity, total eccentricity connectivity, augmented eccentric connectivity, first Zagreb eccentricity, modified eccentric connectivity, second Zagreb eccentricity, and the edge version of eccentric connectivity indices, are computed for the molecular graph of a PolyEThyleneAmidoAmine (PETAA) dendrimer. Moreover, the explicit representations of the polynomials associated with some of these indices are also computed. Full article

Topological indices collect information from the graph of molecule and help to predict properties of the underlying molecule. Zagreb indices are among the most studied topological indices due to their applications in chemistry. In this paper, we compute first and second reverse Zagreb indices, reverse hyper-Zagreb indices and their polynomials of Prophyrin, Propyl ether imine, Zinc Porphyrin and Poly (ethylene amido amine) dendrimers. Full article

Tickysim is a clock tick-based simulator for the inter-chip interconnection network of the SpiNNaker architecture. Network devices such as arbiters, routers, and packet generators store, read, and write forward data through fixed-length FIFO buffers. At each clock tick, every component executes a “read” phase followed by a “write” phase. The structures of any finite graph which represents numerical quantities are known as topological indices. In this paper, we compute degree-based topological indices of the Tickysim SpiNNaker Model ( $TSM$ ) sheet. Full article

Topological indices and connectivity polynomials are invariants of molecular graphs. These invariants have the tendency of predicting the properties of the molecular structures. The honeycomb network structure is an important type of benzene network. In the present article, new topological characterizations of honeycomb networks are given in the form of degree-based descriptors. In particular, we compute Zagreb and Forgotten polynomials and some topological indices such as the hyper-Zagreb index, first and second multiple Zagreb indices and the Forgotten index, F. We, for the first time, determine some regularity indices such as the Albert index, Bell index and $IRM(G)$ index, as well as the F-index of the complement of the honeycomb network and several co-indices related to this network without considering the graph of its complement or even the line graph. These indices are useful for correlating the physio-chemical properties of the honeycomb network. We also give a graph theoretic analysis of some indices against the dimension of this network. Full article

Topological indices are numerical parameters used to study the physical and chemical properties of compounds. In quantitative structure–activity relationship QSARs, topological indices correlate the biological activity of compounds with their physical properties like boiling point, stability, melting point, distortion, and strain energy etc. In this paper, we determined the M-polynomials of the crystallographic structure of the molecules Cu₂O and T_iF₂ [p,q,r]. Then we derived closed formulas for some well-known topological indices using calculus. In the end, we used Maple 15 to plot surfaces associated with the topological indices of Cu₂O and T_iF₂ [p,q,r]. Full article