Aug 2026· Eskişehir Teknik Üniversitesi Bilim ve Teknoloji Dergisi B - Teorik Bilimler· Vol 14, pp. 53-77· 0 citations· 8 references
TL;DR
Wiener-type distance-based topological indices (Wiener, Wiener polarity, hyper-Wiener, Harary, reciprocal complementary Wiener, and terminal Wiener) are discussed and recalculated by considering the odd and even cases of n.
Abstract
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.
Topological indices play a fundamental role in the study of graph structures and their quantitative properties, particularly in applications related to chemistry, network science, and dynamic system analysis. In this work, we study the Reduced Zagreb index, Zagreb coindices, and the PI index of monogenic semigroup graphs, a well-studied class of algebraic graphs with relevance to both pure and applied mathematics. Using the adjacency structure of the graph as the main tool, we derive exact closed-form expressions for these indices and illustrate the results through explicit examples. Furthermore, we obtain explicit parity-dependent closed-form formulas for the reduced first Zagreb index, reduced second Zagreb index, Zagreb coindices and the vertex-PI index of monogenic semigroup graphs. The derivations are based on a neighbourhood-block decomposition of the edge set, which provides a unified computational framework for several families of degree-based topological indices. These results demonstrate how the algebraic structure of a monogenic semigroup determines the behaviour of important graph-theoretical descriptors.
Seda Oğuz Ünal· Balıkesir Üniversitesi Fen B...· 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.
Jayavel Pooja, Venkatesan Muthukumaran, B. Rather et al.· Discrete Mathematics, Algori...· 0 citations
This paper presents 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, which provides a proper model for the highly branched organic compounds, dendrimer, hyperbranched polymers, and functional nanomaterials.
Babysuganya K, Maheswari M, N. A et al.· Adolescência e Saúde· 0 citations
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.
Topological indices are numerical descriptors used in chemical graph theory to characterize the size, branching and connectivity of molecular structures. These descriptors are significantly correlated with a range of physicochemical characteristics and biological activities of a molecular compound. Recently, the M-polynomial approach has been used to represent molecular structures and calculate degree-based topological indices for various graph structures. This study focuses on determining closed-form expressions of the M-polynomial for three types of silicon-carbon structures: SiC3-I[a,b], SiC3-II[a,b] and SiC3-III[a,b], for arbitrary a>1, b≥1. Using the obtained M-polynomial, we calculate nine degree-dependent indices for these silicon-carbide structures. The study also includes visualizations of the computed topological indices and the M-polynomial. Furthermore, we have done a novel comparative analysis among the topological indices for these specific structures. The results obtained in this study provide a mathematical foundation for future researchers in the property prediction of these structures.
Shibsankar Das, Shahzadi Nargis· Scientific Annals of Compute...· 0 citations
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