The GTPase KRAS executes a conformational switch between a GTP-bound active state and a GDP-bound inactive state, a process central to oncogenic signaling. However, the structural basis of this switching at the level of residue-contact organization remains incompletely characterized by traditional binary structural models. Here, we present a statistical-mechanical generalization of the Gaussian Network Model (GNM) by constructing spanning-tree partition functions for residue-contact graphs using the weighted Kirchhoff Laplacian in conjunction with the Matrix-Tree Theorem. Within this framework, the standard GNM is recovered in the high-temperature limit, whereas the present formulation enables a continuous Boltzmann-weighted ensemble analysis. We compute the network free energy , mean contact energy , heat capacity , and thermodynamic entropy across an effective temperature sweep that maps the combinatorial diversity of the contact network, thereby probing the topological landscape rather than structural melting. Differential analysis reveals that KRAS activation reflects a systematic entropy-enthalpy compensation mechanism: the active state incurs a systematic energetic penalty that is offset by a marked gain in conformational entropy , with a free-energy crossover occurring at . Edge marginal inclusion probabilities, obtained via effective-resistance theory, identify Switch I (residues 25-40) as the primary allosteric locus of nucleotide-driven network reorganization. This approach provides a thermodynamically grounded perspective on KRAS allostery, quantitatively demonstrating how network architecture enables functional versatility through entropy-driven conformational flexibility.

Consider a multigraph $G$ whose edges are colored from $[q]$ ($q$-colored graph) and $\alpha=(\alpha_1,\ldots,\alpha_{q}) \in \mathbb{N}^{q}$ (color-constraint). A subgraph $H$ of $G$ is called $\alpha$-colored if $H$ has exactly $\alpha_i$ edges of color $i$ for each $i \in[q]$. In this paper, we focus on $\alpha$-colored arborescences (spanning out-trees) in $q$-colored multidigraphs. We study the decision, counting and search versions of this problem. It is known that the decision and search problems are polynomial-time solvable when $q=2$ [Barahona and Pulleyblank, Discret. Appl. Math. 1987] and that the decision problem is NP-complete when $q$ is arbitrary [Ardra et al., arXiv 2024]. However the complexity status of the problem for fixed $q$ was open for $q > 2$. We solve this problem using an algebraic approach. Given a $q$-colored digraph $G$ and a vertex $s$ in $G$, we construct a symbolic matrix in $q-1$ indeterminates such that the number of $\alpha$-colored arborescences in $G$ rooted at $s$ for all color-constraints $\alpha \in \mathbb{N}^q$ can be read from its determinant polynomial. This result extends Tutte's matrix-tree theorem and gives a polynomial-time algorithm for the counting and decision problems for fixed $q$. We use it to design an algorithm that finds an $\alpha$-colored arborescence when one exists. We also study the weighted variant of the problem and give a polynomial-time algorithm (when $q$ is fixed and weights are polynomially bounded) which finds a minimum weight solution.

A theta curve is a spatial embedding of the [Formula: see text]-graph in the three-sphere, taken up to ambient isotopy. We define the determinant of a theta curve as an integer-valued invariant arising from the first homology of its Klein cover. When a theta curve is simple, containing a constituent unknot, we prove that the determinant of the theta curve is the product of the determinants of the constituent knots. Our proofs are combinatorial, relying on Kirchhoff’s Matrix Tree Theorem and spanning tree enumeration results for symmetric, signed, planar graphs.

The theory of electrical networks, in its current state, covers a number of areas of contemporary mathematics and mathematical physics including the combinatorics of paths, forests and groves on graphs, discrete harmonic analysis, problems of random walks, exactly solved models in statistical mechanics, cluster varieties related to spaces of totally positive matrices, discrete integrable systems, algebraic structures similar to Zamolodchikov's tetrahedron equation, and many others. The main aim of this survey is to present some of these topics, classical and recently discovered ones alike. Bibliography: 114 titles.

ABSTRACT This article investigates the prescribed‐time stability (PS) of complex systems with hybrid time‐varying delays (CSHTD) via a novel control strategy. In order to be more suitable for practical application, the internal delay, the self‐delay, and the neighboring delay of complex systems are distinguished in this article, and these time delays are considered as time‐varying functions. An appropriate control strategy is designed in this study to overcome the effects of the hybrid time‐varying delays. This strategy can make CSHTD achieve stability within any settling time, independent of control parameters and initial conditions. Next, a suitable global Lyapunov function of CSHTD is established, and a sufficient criterion for the PS of CSHTD is given by combining Lyapunov stability theory with Kirchhoff's matrix tree theorem. Finally, the PS of the coupled oscillators model with hybrid time‐varying delays via the control strategy is studied as an application, and the effectiveness of the proposed approach is verified by numerical simulations.

Temperature in a non-equilibrium system is not well defined. If the non-equilibrium dynamics is Lindbladian, it is possible to associate several apparent temperatures with it. However, the apparent temperature governed by the Lindblad equation and its properties as the system approaches a steady state have not been extensively studied. Representing a finite-dimensional quantum system as a graph, we extended Kirchhoff's matrix-tree theorem to a case where the system's graph is not strongly connected and showed that the diagonal and off-diagonal components of a density matrix can be dynamically evolved separately. We discovered that the apparent temperatures of different interaction channels between the system's graph and the environment can be different. It can be equilibrated in each channel if the system has only cycles with length ℓ≤2. In contrast, when the system has a cycle with length ℓ>2, the apparent temperatures of the system and environment are different at a steady state or not equilibrated. Equilibration of an apparent temperature may occur in some channels but not in others and can be undefined in some cases. Generally, an apparent temperature is not transitive because it depends on the interaction channels between the subsystem and its environment. Considering the apparent temperature for each interaction channel, instead of one value of temperature representing the entire system, will provide more insights about the state of a quantum system and potentially lead to more efficient control of quantum systems for many applications in quantum technology. We demonstrated these points in some important finite-dimensional quantum systems pertaining to quantum memory, thermalizing channels in a qutrit and quantum sensing.

Let G be a multidigraph without self-loops. The complex Laplacian matrix of G, denoted by L C ( G ) , is defined in Barik et al. [On singularity and properties of eigenvectors of complex Laplacian matrix of multidigraphs. AKCE Int J Graphs Comb 2023;20(2):125–133. doi: 10.1080/09728600.2023.2234014]. In this article, we consider the complex signless Laplacian matrix of multidigraphs, denoted by Q C ( G ) . For a simple graph, the Matrix-Tree Theorem gives the number of spanning trees in a graph in terms of the principal minors of its Laplacian matrix. That gives a motivation to study the principal minors of the matrices, which has been done for simple graphs, digraphs, signed graphs and mixed graphs. In this article, we provide a combinatorial description of the principal minors and determinants of both L C ( G ) and Q C ( G ) . As an application, a class of multidigraphs is provided whose complex Laplacian spectrum and complex signless Laplacian spectrum are the same. We obtain a necessary and sufficient condition for a multidigraph to be Q C -singular. Further, the eigenvectors of the Q C -singular multidigraphs are studied.

The stability for highly nonlinear impulsive coupled systems (HNICSs) with multi-weights is studied. It should be pointed out that the effects of impulsive disturbances are considered into highly nonlinear coupled systems. For HNICSs, existing impulsive differential inequalities cannot be used to obtain the stability criteria, which motivates us to construct a novel impulsive differential inequality and the inequality generalises the classic impulsive differential inequality to highly nonlinear cases. Furthermore, based on this inequality, Kirchhoff's Matrix Tree theorem in graph theory and the Lyapunov method, stability criteria are derived. Additionally, the coupled impulsive van der Pol-Duffing oscillators are presented. Correspondingly, the validity and practicability of the derived results are indicated with a numerical example.

The number of spanning trees in graphs (networks) is a fundamental invariant that plays a crucial role in measuring the reliability and connectivity of a network. It is particularly significant in various applications, including network design, circuit analysis, and structural stability assessments. In this paper, we derive explicit and simplified formulas for computing the complexity of specific classes of graphs, particularly trapezoidal graphs, using advanced techniques from linear algebra and matrix analysis. By leveraging Kirchhoff's matrix tree theorem and eigenvalue-based formulations, we establish efficient methods for determining the number of spanning trees. Additionally, we explore computational approaches such as Chio’s condensation and Dodgson’s method to enhance the accuracy and efficiency of determinant calculations related to graph Laplacians. The results obtained provide a deeper insight into the structural properties of trapezoidal graphs and their spanning tree enumeration, offering potential applications in combinatorial optimization, network topology analysis, and applied mathematics.

Abstract We develop recursive relations among abundances in an r-process network evolving neutron captures, photodisintegrations and beta decays through the use of the matrix-tree and matrix-forest theorems. Since these theorems are based on results from graph theory, we term the relations the GrRproc (Graphical R-process) relations. We validate the relations by using them to compute r-process abundances in network calculations in different astrophysical environments. We also illustrate how they can be used to follow complex reaction flows quantitatively in an evolving r-process network through the concept of contribution paths. Such contribution paths show how particular reactions govern the evolution of abundance features during the nucleosynthesis and, consequently, can clarify the role of key nuclear data and astrophysical environments in that evolution. The Python open-source package that implements the tool is freely available.

In discrete choice experiments, the information matrix depends on the model parameters. Therefore designing optimally informative experiments for arbitrary initial parameters often yields highly nonlinear optimization problems and makes optimal design infeasible. To overcome such challenges, we connect design theory for discrete choice experiments with Laplacian matrices of undirected graphs, resulting in complexity reduction and feasibility of optimal design. We rewrite the D-optimality criterion in terms of Laplacians via Kirchhoff’s matrix tree theorem, and show that its dual has a simple description via the Cayley–Menger determinant of the Farris transform of the Laplacian matrix. This results in a drastic reduction of complexity and allows us to implement a gradient descent algorithm to find locally D-optimal designs. For the subclass of Bradley–Terry paired comparison models, we find a direct link to maximum likelihood estimation for Laplacian-constrained Gaussian graphical models. Finally, we study the performance of our algorithm and demonstrate its application to real and simulated data.

The linear framework is an approach to analysing biochemical systems based on directed graphs with labelled edges. When applied to individual molecular systems, graph vertices correspond to system states, directed edges to transitions, and edge labels to transition rates. Such a graph specifies the infinitesimal generator of a continuous-time Markov process. The master equation of this Markov process, which describes the forward evolution of vertex probabilities, is a linear differential equation, after which the framework is named, whose operator is the Laplacian matrix of the graph. The Matrix-Tree theorem, when applied to this Laplacian matrix, allows the steady-state probabilities of the Markov process to be expressed as rational algebraic functions of the transition rates. This capability gives algebraic access to problems that have otherwise been treated by approximations or numerical simulations, and enables theorems to be proved about biochemical systems that rise above their underlying molecular complexity. Here, we extend this capability from the steady state to the transient regime. We use the All-Minors Matrix-Tree theorem to express the moments of the conditional first-passage time distribution, and the corresponding splitting probabilities, as rational algebraic functions of the transition rates. This extended capability brings many new biological problems within the scope of the linear framework.

This paper is intended to study noise-to-state stability in probability (NSSP) for random coupled Kuramoto oscillators with input control (RCKOIC). A feedback control is designed, which makes us give the existence and uniqueness of a solution for RCKOIC. Based on Kirchhoff’s matrix tree theorem in graph theory, an original and appropriate Lyapunov function for RCKOIC is established. With the help of the Lyapunov method and by resorting to some analysis skills, NSSP for RCKOIC with an arbitrarily coupled topological structure and second-order moment process stochastic disturbance is acquired. Finally, the effectiveness of the obtained results is verified by a numerical test and its simulation process.

For a bipartite graph $G$ on parts of cardinality $m$ and $n$, the bipartite complement of $G$ is defined as the graph obtained from the complete bipartite graph $K_{m,n}$ by removing the edges of an isomorph of $G$. In this paper, based on the matrix-tree theorem and the Schur complement technique, we give a determinant expression for the number of spanning trees of the bipartite complement of a semiregular bipartite graph. As by-products, we show that the corresponding result can generalize some previous results on the problem of enumerating spanning trees and obtain an explicit formula for the number of spanning trees of the bipartite complement of the subdivison of a regular circulant graph.

This paper investigates the stability of predator–prey models within multi-patch environments, with a particular focus on the influence of cross-dispersion across patches. We apply Kirchhoff’s matrix tree theorem and Liapunov’s method to derive criteria related to the cross-dispersion topology, thus solving the challenge of determining global asymptotic stability conditions. The method incorporates realistic ecological interactions and spatial heterogeneity, offering a framework for stability analysis. Our findings demonstrate that an appropriate level of cross-dispersion can effectively mitigate oscillations and foster convergence toward equilibrium. Two numerical examples validate these theoretical results and demonstrate the feasibility and effectiveness of the model across multiple patches.

An age-structured mathematical model for studying Malaria transmission dynamics: Applications to some areas of Senegal

Living systems are maintained out of equilibrium by external driving forces. At stationarity, they exhibit emergent selection phenomena that break equilibrium symmetries and originate from the expansion of the accessible chemical space due to nonequilibrium conditions. Here, we use the matrix-tree theorem to derive upper and lower thermodynamic bounds on these symmetry-breaking features in linear and catalytic biochemical systems. Our bounds are independent of the kinetics and hold for both closed and open reaction networks. We also extend our results to master equations in the chemical space. Using our framework, we recover the thermodynamic constraints in kinetic proofreading. Finally, we show that the contrast of reaction-diffusion patterns can be bounded only by the nonequilibrium driving force. Our results provide a general framework for understanding the role of nonequilibrium conditions in shaping the steady-state properties of biochemical systems.

Generating functions and counting formulas for spanning trees and forests in hypergraphs

<abstract><p>Stochastic complex networks with multi-weights which were driven by Brownian motion were widely investigated by many researchers. However, Brownian motion is not suitable for the modeling of engineering issues by reason of its variance, which is infinite at any time. So, in this paper, a novel kind of stochastic complex network with multi-weights driven by second-order process is developed. To disclose how the weights and second-order process affect the dynamical properties of stochastic complex networks with multi-weights driven by the second-order process, we discuss exponential stability of the system. Two types of sufficient criteria are provided to ascertain exponential stability of the system on the basis of Kirchhoff's matrix tree theorem and the Lyapunov method. Finally, some numerical examples are given to verify the correctness and validity of our results.</p></abstract>

As we all know, the number of spanning trees can be calculated by virtue of the famous Kirchhoff’s matrix-tree theorem. However, it doesn’t work when coming across complex graphs with thousands of edges and vertices or more. Hence, how to obtain accurate solutions of the number of spanning trees of general fan graphs becomes a subject to several studies of many places like computer science, physics and mathematics. In this paper, we focus on calculating different types of graphs generated by adding vertices and edges to the fan graph. Particularly, we define a new graph called "C-graph", which brings a unique angle of view for us to recognize the construction of original graphs. Moreover, we introduce a new iterative relation about the general fan graph, simplifying the calculation. Therefore, we can obtain functions of the number of spanning trees obtained from given fan graphs, which are also suitable for larger and more complex conditions. Finally, we discuss the effect of vertices and edges on the number of spanning trees, finding out that edges have greater impact. Additionally, by using Kirchhoff’s matrix-tree theorem, we verified the rationality of our results.