Jun 2026· Journal of Science and Arts· Vol 26, pp. 357· 0 citations· 5 references
Abstract
Graphs are fundamental structures widely used across diverse scientific fields, including chemistry, biology, network theory, and the social sciences. In recent years, the study of graph-theoretic models involving specific integer sequences has attracted considerable interest, owing to their rich combinatorial and algebraic characteristics. Among these sequences, the Tetranacci numbers stand out for their notable connections to continued fractions, quadratic fields, and certain classes of Diophantine equations. In this study, we introduce a new family of graphs, termed “Tetranacci graphs”, whose degree sequences are composed of n consecutive Tetranacci numbers. Using the graph invariant Ω(D) — a tool that provides critical insights into the structural features of graphs such as realizability, connectivity, and cyclic components— we explore the conditions under which such sequences yield valid graphical realizations. We derive necessary and sufficient criteria for the realizability of these sequences for any positive integer n, and we classify all possible graphical forms (Tetranacci graphs) explicitly for the cases 1≤n≤4. Our analysis is further extended to offer a general structural characterization for the case n ≥5. These fundamental criteria establish a comprehensive framework for studying Tetranacci graphs, opening new avenues for research into the interplay between number theory and graph realizability.
Finite groups are rigid algebraic objects, whose Cayley graphs expose a rich network geometry through which group-theoretic structure can be measured, compared, and learned. In this paper, a dataset of $131{,}406$ Cayley graphs is constructed, covering all groups of order at most $767$ except order $512$, recording exact algebraic labels for group properties together with a broad collection of graph, cycle, distance, and spectral statistics. This census aims to provide novel benchmarks for studying how finite-group properties are reflected in Cayley graph observables. It also yields new enumerative contributions: alongside recovering known OEIS sequences for standard group classes, new sequences for monolithic groups and for groups generated by at most three, four, and five elements are contributed to the OEIS. The accompanying network analysis identifies several empirical regularities and formulates testable conjectures, including relationships involving square clustering, Cayley graph diameter, average graph disorder, and spectral eigengaps of nilpotent groups. Finally, a comparison between classical models, an MLP, and graph neural network architectures is performed for predicting algebraic group properties directly from Cayley graph data. The results show that engineered graph statistics are highly informative, while GNNs, especially GIN and in some fixed-order settings GCN, can recover substantial structural signal directly from the graph. Such that graph-aware architectures show phases of optimality on these group-theoretic graph representations.
Rashid Barket, Enrico Grimaldi, Yacoub Hendi et al.· 0 citations
Graphons are measurable functions used to describe the asymptotic behavior of convergent graph families. Originally motivated by problems in combinatorics and graph theory, graphons have found numerous applications in the modeling and analysis of dynamical processes on networks. In this work, we use graphons to formulate the Ising model on convergent graph sequences, which include many network topologies common in applications. We derive the mean-field limit for the resulting model and obtain exact results for phase transitions in such systems. Specifically, we show that the critical temperatures of the Ising model on graphons are determined by the eigenvalues of the Hilbert-Schmidt operator associated with the graph limit. For many important network topologies, these eigenvalues can be computed explicitly. We illustrate our results with three representative random network models: Erd\H{o}s-R\'{e}nyi, small-world, and power-law. In the small-world case, we demonstrate phase transitions to both ferromagnetic and antiferromagnetic phases, as well as coexistence of local minima of the free energy. The latter gives rise to multistability, as confirmed by Monte Carlo simulations. The results of this work demonstrate that the Ising model on graphons combines the analytical tractability of exactly solvable mean-field models with the ability to accommodate a broad range of network topologies. We expect that the use of graphons in spin models will lead to new insights into the statistical physics of interacting systems on complex networks.
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
Let G be a finite group. We introduce a new graph definition, called the pseudo A4-graph, as an extension of the classical A4-graph. The pseudo A4-graph of G or PA4-graph is a simple graph ΓG whose vertices are elements of G, and two distinct vertices a and b are joined by an edge if and only if ab−1 = ba−1. This paper presents the formulas for certain graph invariants such as the number of edges, diameter, total degree, chromatic number, clique number, independence number, domination number, matching number, and graph energy. The method involves constructing the A4-graphs of dihedral groups, analyzing their structure, and systematically computing graph invariants through algebraic reasoning and combinatorial techniques to establish fundamental properties and relationships. One of the results in this paper comply with the well-known fact that the energy of a graph is always an even integer.
S. M. Kasim, S. Husain· Journal of Physics, Conferen...· 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.