Jul 2026· Scientific Annals of Computer Science· Vol 36, pp. 89· 0 citations· 25 references
TL;DR
A novel extension of the spectral invariant called Second Hyper Zagreb spectral radius is obtained by replacing the adjacency matrix with the Second Hyper Zagreb matrix and the two graph operations that are primarily covered are splitting and shadow graphs.
Abstract
Chemical graph theory is essential for deriving graph spectral radius, especially in quantum chemistry; a significant link exists between eigenvalues and this invariant of spectral graph theory. This invariant is derived from spectrum, and has numerous applications in various fields and is used to solve many real-life problems. Zagreb-type indices and their spectrum variants are crucial for QSAR/QSPR modeling, chemoinformatics, and computational drug design [16]. The literature contains numerous variants of graph spectral radius. In this article, we concentrate on a novel extension of the spectral invariant called Second Hyper Zagreb spectral radius, which is obtained by replacing the adjacency matrix with the Second Hyper Zagreb matrix. The two graph operations that are primarily covered in this article are splitting and shadow graphs. The connection between the Second Hyper Zagreb spectral radius of these graph operations and Second Hyper Zagreb spectral radius of base graph Gis of special relevance to us. By using these operations, we can address the challenge of figuring out the relationship between the spectral radius of the base graph Gand the spectral radius of the newly generated graph. The findings of this article improve our understanding and pave the way for future research in spectral graph theory, where graph operations are frequently employed and play an important role in solving real-world problems.
This study explores the spectral characteristics and energy distributions associated with selected graph operations derived from the first Zagreb, second Zagreb, and sum-connectivity matrices to contribute to understanding how algebraic operations induce spectral energy shifts analogous to perturbations in physical or molecular graph systems.
S. Sripriya, A. Anuradha· Baghdad Science Journal· 0 citations
This thesis investigates two central directions in algebraic graph theory, with an emphasis on spectral methods: spectral determination of graphs and transitivity properties of generalized-Hamming graphs and their complements. The first part focuses on graphs that are determined by the spectra of associated matrices. We study spectral determination with respect to the adjacency, Laplacian, signless Laplacian, and normalized Laplacian matrices, with particular emphasis on the adjacency spectrum. We survey existing results on graphs determined by their spectrum and develop new proof techniques for establishing spectral uniqueness. In particular, we present new proofs for the spectral characterization of complete bipartite graphs and Tur\'{a}n graphs, as well as some new results related to the spectral characterization of the important family of strongly regular graphs. In addition, we introduce a new family of graphs, called \emph{the graphs of pyramids}, and prove that they are determined by their adjacency spectrum using tools from matrix analysis, such as Cauchy's interlacing theorem and Schur complements. The second part of the thesis studies generalized-Hamming graphs, a family of Cayley graphs that generalize the sub-family of Hamming graphs, and their complements. We classify the parameters for which these graphs are edge-transitive or even distance-transitive. Our analysis combines spectral methods, group-theoretic arguments, and techniques from the theory of association schemes. As an application, we derive closed-form expressions for the Lov\'{a}sz $\vartheta$-function of generalized-Hamming graphs and their complements whenever either the graph or its complement is edge-transitive. Overall, the results demonstrate how spectral methods provide powerful tools for understanding the structure and symmetry of graphs, and they suggest several directions for further research.
The topological indices are fundamental tools in chemical graph theory with the aim of numerically describing the structural properties of molecular graphs and also predicting physicochemical and biological parameters. In this paper, we present a new graph family called Double Star Double Fanbell (BSDF) which consists of double star graph with each pendant vertex added with a double fan graph. Eight important distance based topological indices are considered, namely Wiener index, Hyper-Wiener index, Harary index, Reciprocal Complementary Wiener index, Wiener Polarity index, Terminal Wiener index, Reverse Wiener index, and Reciprocal Reverse Wiener index are considered and exact closed-form expressions are obtained. The analytical formulations are derived by a systematic distance-partitioning method and written in terms of the graph parameters, which allows the formula to be computed efficiently for any graph size without a need for any repeated shortest-path computations.
In addition to their mathematical importance, the calculated topological descriptors are useful molecular descriptors in Chemical Graph Theory, in which atoms are depicted as vertices, and chemical bonds as edges. These descriptors can be used effectively in Quantitative Structure–Property Relationship (QSPR) and Quantitative Structure–Activity Relationship (QSAR) models to predict molecular stability, boiling point, melting point, solubility, lipophilicity, biological activity, toxicity and pharmacokinetic properties. Moreover, the proposed BSDF graph provides a proper model for the highly branched organic compounds, dendrimer, hyperbranched polymers, and functional nanomaterials. Expressions obtained in this work are in an exact form, which is very appealing for large scale molecular databases, virtual screening, cheminformatics and AI-assisted drug discovery. Therefore, the suggested graph is not only playing a theoretical role in advancement of graph theory but also in the present day computational chemistry and pharmaceutical research.
Babysuganya K, Maheswari M, N. A et al.· Adolescência e Saúde· 0 citations
Topological indices can be used to characterize molecular topology. These are numerical measurements of a suggested molecule’s fundamental structural characteristics, derived from its molecular structure. This numerical value, which is derived from a chemical configuration, represents the important physical properties of the proposed molecule. We use an algebraic number to connect the chemical composition with various physical characteristics, biological activity, and chemical reactivity. A graph with a vertex set of [Formula: see text] in which two unique vertices are adjacent when one element is an integral power of the other is called a power graph [Formula: see text] of a finite group [Formula: see text]. This paper investigates several types of topological indices of power graphs for different finite groups based on distance, degree, and independent sets. We compute the Hyper Wiener Index, Degree Distance Index, Additively Weighted Harary Index, Gutman Index, Multiplicatively Weighted Harary Index, Eccentric Connectivity Index, Connective Eccentricity Index, and Merrifield Simmons Index of power graphs for finite cyclic and non-cyclic groups of order [Formula: see text], dihedral and generalized quaternion groups, where [Formula: see text] are distinct primes. As a consequence, we fix the flaws in the results of [5] and present their correct forms.
Topological indices are mathematical tools that numerically express the topological properties of molecular structures that can be represented by graphs. These indices are widely used in various disciplines such as biology, computer science, and network theory, as well as chemistry. In graph theory, many topological indices have been defined to measure different topological properties. However, explicit formulations of certain Wiener-type distance-based indices that take into account the odd–even structure of the number of vertices and their behavior on some special unicyclic graph families remain limited in the literature. In this work, Wiener-type distance-based topological indices (Wiener, Wiener polarity, hyper-Wiener, Harary, reciprocal complementary Wiener, and terminal Wiener) are discussed. First, these indices for the well-known cycle graph C_n and the path graph P_n are recalculated by considering the odd and even cases of n. Then, these Wiener-type indices for the turnip graph, lollipop graph, and sun graph, which are well-known unicyclic graphs, are calculated and presented depending on the graph parameters and the odd–even status of n. The results obtained in this study extend the existing results in the literature by providing explicit expressions for these indices under parity conditions and for specific unicyclic graph structures. Therefore, this work contributes to a better understanding and comparison of the topological properties of these graph families and provides a useful basis for future studies on topological index calculations for graphs with similar structures.
H. Topcu, Eda Güner· Eskişehir Teknik Üniversites...· 0 citations
The stress of a graph is the number of geodesics (shortest paths) that pass through each vertex. A topological index of a chemical structure (graph) is a number that correlates the chemical structure with its chemical reactivity or physical properties. In this paper, we introduce a new topological
index for graphs called the second hyper-Gourava stress index, which is defined using the stresses of vertices. Further, we establish several inequalities, prove related results, and compute the second hyper-Gourava stress index for some standard graphs. Similarly, we establish the importance of the second hyper-Gourava stress index in predicting the physicochemical properties of
lower alkanes.
Girija K. P., Ravi Kumar C. K, Rahul Munavalli et al.· Global and Stochastic Analys...· 0 citations