Search Results (29)

Zagreb-type indices are topological indices derived from the degrees of nodes. The first Zagreb index, the F-index, and the Y-index represent the sum of the squares, cubes, and fourth powers of all node degrees, respectively. These indices are valuable for understanding the chemical reactions, physical characteristics, and biological activities of various substances. In this study, we explore the connection between Y-index and the graph Laplacian spectrum. Additionally, we introduce the fractal graphs based on star graphs, a class of extended Vicsek graphs, and derive the rules for eigenvalue evolution between two generations of the graph. Ultimately, we provide exact closed-form expressions for the first Zagreb index, F-index, and Y-index of the fractal graphs based on star graphs by using spectral graph theory. Full article

In this article, we introduce and investigate a novel class of graphs that are called Power Congruence Graph PCGs, which are defined over the vertex set V $=\{0,1,2,\dots ,n-1\}$ where two vertices $a,b\in V$ are adjacent if $a^k\equiv b^k(modm)$ for some modulus $m\in M_p$, where $M_p=\{p,p^2,\dots ,p^t\mid p^t<n\}$. We thoroughly characterize the structural features of these graphs, establishing that each PCG decomposes into a union of $d+1$ complete components, where $d={\displaystyle \frac{p-1}{gcd(k,p-1)}}$. The component sizes are explicitly given for n, p, and k. This decomposition highlights symmetry patterns in the component arrangement, emphasizing connectedness and structural balance. We derive key graph-theoretic metrics such as degree distribution, size, chromatic number, clique number and domination number. We also compute the adjacency and Laplacian matrices, as well as their spectra and associated graph energies to better understand the structural similarities and differences among PCGs with different exponents and prime moduli. This paper offers a systematic framework for comprehending power congruence based graph constructs, integrating number theory with structural and spectral graph theory and illustrating the natural symmetry that underpins these combinatorial structures. Full article

This paper establishes a graph-theoretic framework for idealization semigroups arising from Bruck–Reilly extensions. Building on a recent study by Wazzan and Ozalan, we introduce five graph families—${\Gamma }_E$, ${\Gamma }_0$, ${\Gamma }_{\mathrm{Cay}}$, ${\Gamma }_{\mathcal{K}}$, and $\Gamma \left(G_k\right)$—each encoding a distinct algebraic facet of $S{B}_i(\bowtie )B$. We prove explicit correspondences linking combinatorial invariants to algebraic structure: diameter captures generating efficiency and semilattice height; girth signals short relations; chromatic number bounds idempotent cardinalities and $\mathcal{D}$-class counts; clique number measures maximal commuting subsets; and Laplacian spectra encode ideal size and Schützenberger groups. Our central result demonstrates that Green’s relations are combinatorially recoverable from graph pairs. For commutative $S{B}_i(\bowtie )B$, $({\Gamma }_E,{\Gamma }_{\mathcal{K}})$ uniquely determines $\mathcal{J}$-order, $\mathcal{D}$-classes, and $\mathcal{H}$-classes via neighborhood inclusions, bipartite components, and automorphism orbits, yielding the first algorithmic reconstruction of ideal-theoretic structure from graph data. The framework is implemented in SageMath as a reproducible open-source toolkit validated on concrete examples. This work synthesizes algebraic graph theory, semigroup theory, and computational mathematics into a unified algebraic-combinatorial dictionary, providing both new analytical tools and a methodological template for studying algebraic constructions via graph invariants. Full article

This study analyzes network coherence in two-layer and three-layer networks with positive and negative inter-layer correlation patterns. Using algebraic graph theory, we construct the Laplacian matrices for different correlated graph structures and compute their Laplacian spectra. The impact of correlation patterns on network coherence is investigated for two-layer and three-layer structures, and numerical evaluations are performed. The results show that negative-correlation patterns yield better network coherence than positive ones. This work provides fundamental insights into the structural robustness of layered networks and offers theoretical guidance for the design of robust networked systems. Full article

This paper extends the spectral analysis of distance-based matrices associated with chemical graph structures of order n, focusing on the distance matrix $\mathcal{D}\left(\mathcal{G}\right)$, the distance Laplacian $\mathcal{DL}\left(\mathcal{G}\right)$, and the distance signless Laplacian $\mathcal{DQ}\left(\mathcal{G}\right)$. We investigate the spectral integrality of these matrices for selected acyclic and cyclic hydrocarbon molecular graphs by examining whether their corresponding spectra consist entirely of integers. In addition, we compute and compare the associated distance energies, namely, the distance energy ${\mathcal{E}}_{\mathcal{D}}$, the distance Laplacian energy ${\mathcal{E}}_{\mathcal{DL}}$, and the distance signless Laplacian energy ${\mathcal{E}}_{\mathcal{DQ}}$ to explore their structural significance. Using computational tools, we present numerical results and graphical comparisons that reveal meaningful relationships among these energies. In particular, our analysis establishes the conjecture in the form of a strict inequality ${\mathcal{E}}_{\mathcal{DL}}>{\mathcal{E}}_{\mathcal{DQ}}>{\mathcal{E}}_{\mathcal{D}}$. These findings demonstrate that the distance Laplacian energy is more sensitive to molecular structural variations, highlighting its effectiveness as a discriminative molecular descriptor in chemical graph theory. Full article

The Clausius–Mossotti (CM) factor underpins the theoretical description of dielectrophoresis (DEP) and is widely used in micro- and nano-scale systems for frequency-dependent particle and cell manipulation. It has further been proposed as an “electrophysiology Rosetta Stone” capable of linking DEP spectra to intrinsic cellular electrical properties. In this paper, the mathematical foundations and interpretive limits of this proposal are critically examined. By analyzing contrast factors derived from Laplace’s equation across multiple physical domains, it is shown that the CM functional form is a universal consequence of geometry, material contrast, and boundary conditions in linear Laplacian fields, rather than a feature unique to biological systems. Key modelling assumptions relevant to DEP are reassessed. Deviations from spherical symmetry lead naturally to tensorial contrast factors through geometry-dependent depolarisation coefficients. Complex, frequency-dependent CM factors and associated relaxation times are shown to inevitably arise from the coexistence of dissipative and storage mechanisms under time-varying forcing, independent of particle composition. Membrane surface charge influences DEP response through modified interfacial boundary conditions and effective transport parameters, rather than by introducing an independent driving mechanism. These results indicate that DEP spectra primarily reflect boundary-controlled field–particle coupling. From an inverse-problem perspective, this places fundamental constraints on parameter identifiability in DEP-based characterization. The CM factor remains a powerful and general modelling tool for micromachines and microfluidic systems, but its interpretive scope must be understood within the limits imposed by Laplacian field theory. Full article

We construct a family of self-adjoint operators on the prime numbers whose entries depend on pairwise arithmetic divergences, replacing geometric distance with number-theoretic dissimilarity. The resulting spectra encode how coherence propagates through the prime sequence and define an emergent arithmetic geometry. From these spectra we extract observables such as the heat trace, entropy, and eigenvalue growth, which reveal persistent spectral compression): eigenvalues grow sublinearly, entropy scales slowly, and the inferred dimension remains strictly below one. This rigidity appears across logarithmic, entropic, and fractal-type kernels, reflecting intrinsic arithmetic constraints. Analytically, we show that for the unnormalized Laplacian, the continuum limit of its squared Hamiltonian corresponds to the one-dimensional bi-Laplacian, whose heat trace follows a short-time scaling proportional to $t^{-1/4}$. Under the spectral dimension convention $d_s=-2d\log \Theta /d\log t$, this result produces $d_s=1/2$ directly from first principles, without fitting or external hypotheses. This value signifies maximal spectral compression and the absence of classical diffusion, indicating that arithmetic sparsity enforces a coherence-limited, non-Euclidean geometry linking spectral and number-theoretic structure. Full article

Cayley graphs sit at the intersection of algebra, geometry, and theoretical computer science. Their spectra encode fine structural information about both the underlying group and the graph itself. Building on classical work of Alon–Milman, Dodziuk, Margulis, Lubotzky–Phillips–Sarnak, and many others, we develop a unified representation-theoretic framework that yields several new results. We establish a monotonicity principle showing that the algebraic connectivity never decreases when generators are added. We provide closed-form spectra for canonical 3-regular dihedral Cayley graphs, with exact spectral gaps. We prove a quantitative obstruction demonstrating that bounded-degree Cayley graphs of groups with growing abelian quotients cannot form expander families. In addition, we present two universal comparison theorems: one for quotients and one for direct products of groups. We also derive explicit eigenvalue formulas for class-sum-generating sets together with a Hoffman-type second-moment bound for all Cayley graphs. We also establish an exact relation between the Laplacian spectra of a Cayley graph and its complement, giving a closed-form expression for the complementary spectral gap. These results give new tools for deciding when a given family of Cayley graphs can or cannot expand, sharpening and extending several classical criteria. Full article

Let $\mathfrak{R}$ be a commutative ring with identity $1\ne 0$. The weakly zero-divisor graph of $\mathfrak{R}$, denoted $W_{\Gamma }(\mathfrak{R})$, is the simple undirected graph whose vertex set consists of the nonzero zero-divisors of $\mathfrak{R}$, where two distinct vertices $\mathfrak{a}$ and $\mathfrak{b}$ are adjacent if and only if there exist $r\in \mathrm{ann}(\mathfrak{a})$ and $s\in \mathrm{ann}(\mathfrak{b})$ such that $rs=0$. In this paper, we study the signless Laplacian spectrum of $W_{\Gamma }({\mathbb{Z}}_n)$ for several composite forms of n, including $n=p^2{q}^2$, $n=p^{2}qr$, $n=p^m{q}^m$ and $n=p^{m}qr$, where $p, q, r$ are distinct primes and $m\ge 2$. By using generalized join decomposition and quotient matrix methods, we obtain explicit eigenvalue formulas for each case, along with structural bounds, spectral integrality conditions and Nordhaus–Gaddum-type inequalities. Illustrative examples with computed spectra are provided to validate the theoretical results, demonstrating the interplay between the algebraic structure of ${\mathbb{Z}}_n$ and the spectral properties of its weakly zero-divisor graph. Full article

Persistent topological Laplacians constitute a new class of tools in topological data analysis (TDA). They are motivated by the necessity to address challenges encountered in persistent homology when handling complex data. These Laplacians combine multiscale analysis with topological techniques to characterize the topological and geometrical features of functions and data. Their kernels fully retrieve the topological invariants of corresponding persistent homology, while their non-harmonic spectra provide supplementary information. Persistent topological Laplacians have demonstrated superior performance over persistent homology in the analysis of large-scale protein engineering datasets. In this survey, we offer a pedagogical review of persistent topological Laplacians formulated in various mathematical settings, including simplicial complexes, path complexes, flag complexes, digraphs, hypergraphs, hyperdigraphs, cellular sheaves, and N-chain complexes. Full article

In this paper, three sorts of network indices for the weighted three-layered graph are studied through the methods of graph spectra theory combined with analysis methods. The concept of union of graphs are applied to design two sorts of weighted layered multi-fan composed graphs, and the accurate mathematical expressions of the network indices are obtained through the derivations of Laplacian spectra; furthermore, the asymptotic properties are also derived. We find that when the cardinalities of the vertices on a sector-edge-link tend to infinity, the indices of FONC and EMFPT are irrelevant with the number of copies of the fan-substructure based on the considered graph framework. Full article

In this article, a robust index named first-order network coherence (FONC) for the multi-agent systems (MASs) with layered lattice-like structure is studied via the angle of the graph spectra theory. The union operation of graphs is utilized to construct two pairs of non-isomorphic layered lattice-like structures, and the expression of the index is acquired by the approach of Laplacian spectra, then the corresponding asymptotic results are obtained. It is found that when the cardinality of the node sets of coronary substructures with better connectedness tends to infinity, the FONC of the whole network will have the same asymptotic behavior with the central lattice-like structure in the considered classic graph frameworks. The indices of the networks were simulated to illustrate the the asymptotic results, as described in the last section. Full article

Graph distance measures have emerged as an effective tool for evaluating the similarity or dissimilarity between graphs. Recently, there has been a growing trend in the application of movie networks to analyze and understand movie stories. Previous studies focused on computing the distance between individual characters in narratives and identifying the most important ones. Unlike previous techniques, which often relied on representing movie stories through single-layer networks based on characters or keywords, a new multilayer network model was developed to allow a more comprehensive representation of movie stories, including character, keyword, and location aspects. To assess the similarities among movie stories, we propose a methodology that utilizes a multilayer network model and layer-to-layer distance measures. We aim to quantify the similarity between movie networks by verifying two aspects: (i) regarding many components of the movie story and (ii) quantifying the distance between their corresponding movie networks. We tend to explore how five graph distance measures reveal the similarity between movie stories in two aspects: (i) finding the order of similarity among movies within the same genre, and (ii) classifying movie stories based on genre. We select movies from various genres: sci-fi, horror, romance, and comedy. We extract movie stories from movie scripts regarding character, keyword, and location entities to perform this. Then, we compute the distance between movie networks using different methods, such as the network portrait divergence, the network Laplacian spectra descriptor (NetLSD), the network embedding as matrix factorization (NetMF), the Laplacian spectra, and D-measure. The study shows the effectiveness of different methods for identifying similarities among various genres and classifying movies across different genres. The results suggest that the efficiency of an approach on a specific network type depends on its capacity to capture the inherent network structure of that type. We propose incorporating the approach into movie recommendation systems. Full article

For a finite commutative ring $\mathfrak{R}$ with identity $1\ne 0$, the weakly zero-divisor graph of $\mathfrak{R}$ denoted as $W\Gamma \left(\mathfrak{R}\right)$ is a simple undirected graph having vertex set as a set of non-zero zero-divisors of $\mathfrak{R}$ and two distinct vertices $\mathfrak{a}$ and $\mathfrak{b}$ are adjacent if and only if there exist elements $r\in \mathrm{ann}\left(\mathfrak{a}\right)$ and $s\in \mathrm{ann}\left(\mathfrak{b}\right)$ satisfying the condition $rs=0$. The zero-divisor graph of a ring is a spanning sub-graph of the weakly zero-divisor graph. This article finds the normalized Laplacian spectra of the weakly zero-divisor graph $W\Gamma \left(\mathfrak{R}\right)$. Specifically, the investigation is carried out on the weakly zero-divisor graph $W\Gamma \left({\mathbb{Z}}_{\mathfrak{n}}\right)$ for various values of $\mathfrak{n}$. Full article

An important challenge in the study of complex systems is to identify appropriate effective variables at different times. In this paper, we explain why structures that are persistent with respect to changes in length and time scales are proper effective variables, and illustrate how persistent structures can be identified from the spectra and Fiedler vector of the graph Laplacian at different stages of the topological data analysis (TDA) filtration process for twelve toy models. We then investigated four market crashes, three of which were related to the COVID-19 pandemic. In all four crashes, a persistent gap opens up in the Laplacian spectra when we go from a normal phase to a crash phase. In the crash phase, the persistent structure associated with the gap remains distinguishable up to a characteristic length scale ${ϵ}^{*}$ where the first non-zero Laplacian eigenvalue changes most rapidly. Before ${ϵ}^{*}$, the distribution of components in the Fiedler vector is predominantly bi-modal, and this distribution becomes uni-modal after ${ϵ}^{*}.$ Our findings hint at the possibility of understanding market crashs in terms of both continuous and discontinuous changes. Beyond the graph Laplacian, we can also employ Hodge Laplacians of higher order for future research. Full article