Search Results (61)

Binding affinity prediction in computational drug discovery is confounded by trivial correlations between molecular size and measured potency. We introduce cellular sheaf Laplacians as descriptors of ligand molecular geometry that quantify geometric frustration independent of system size. Sheaves are constructed over molecular graphs by assigning three-dimensional coordinate spaces to atoms and projection operators encoding ideal bonding geometry to edges; eigendecomposition of the resulting Laplacian yields spectral features measuring inconsistencies between local geometric constraints and global topology. Applied to 14,050 protein-ligand complexes from the PDBbind v2020 refined set, MW-residualized Sheaf features capture a statistically significant geometric signal ($r_{\mathrm{partial}} = 0.171$, $p<{10}^{-70}$) that is orthogonal to the Wiener index ($r=0.013$) and persists after controlling for both molecular weight and classical graph-theoretic descriptors ($r_{\mathrm{partial}} = 0.390$, $p<{10}^{-9}$). Sheaf spectral features alone achieve predictive performance ($R^2=0.403$) approaching that of fourteen classical cheminformatics descriptors ($R^2=0.446$), and their combination yields consistent improvements across the binding affinity spectrum (RMSE $=1.43$$p{K}_d$). Permutation importance analysis confirms the Sheaf Frobenius norm as the second most influential descriptor after molecular weight. We introduce Topological Binding Efficiency as a size-normalized quality metric identifying ligands that achieve potent binding through geometric complementarity rather than molecular bulk. Gaussian mixture analysis of the maximum eigenvalue distribution among strong binders reveals two distinct spectral modes corresponding to planar aromatic and three-dimensional sp3-rich scaffolds, confirmed by significant differences in fraction of sp3 carbons and aromatic ring counts ($p<{10}^{-8}$). As an intentionally ligand-centric framework, our approach complements rather than replaces protein-aware co-modelling architectures. This work establishes cellular sheaf theory as a principled framework for encoding molecular topology with statistically significant associations with binding affinity, providing interpretable geometric insights that are inaccessible to conventional molecular descriptors. Full article

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

A graph is planar if it can be embedded in the plane such that its edges intersect only at their common endpoints. In this paper, we determine the graphs attaining the second and third largest signless Laplacian spectral radii among all planar graphs of order $n\ge 398$. Furthermore, we apply this characterization to identify the planar graphs that achieve the first three largest values of the sum of the first and second largest signless Laplacian eigenvalues. Full article

For network graphs, numerous graph features are intimately linked to eigenvalues of the Laplacian matrix, such as connectivity and diameter. Thus, it is very important to solve eigenvalues of the Laplacian matrix for graphs. Similarly, for higher-order networks, eigenvalues of combinatorial Laplacian matrices are also important for invariants of graphs. However, for large-scale networks, it is difficult to calculate eigenvalues of the Laplacian matrix directly because it is either very difficult to obtain the whole network structure or requires a lot of computing resources. Therefore, this article makes the following contributions. Firstly, this paper proposes a random walk approach for estimating the bounds of the greatest eigenvalues of Laplacian matrices for large-scale networks. Considering the relationship between the spectral moments of the adjacency matrix and the closed paths in the network, we utilize the relationship between the adjacency matrix and the Laplacian matrix to establish the relationship between the Laplacian matrix and the closed paths. Then, we employ equiprobable random walks to sample the large graph to obtain the small graph. Through algebraic topology knowledge, we obtain the bounds of the largest eigenvalue of the Laplacian matrix of the large graph by using Laplacian spectral moments of the small graph. Secondly, for high-order networks, this paper proposes a method based on random walks and graph transformations. The graph transformation we propose mainly converts graphs with second-order simplices into ordinary weighted graphs, thereby transforming the problem of solving the spectral moments of the second-order combined Laplacian matrix into solving the spectral moments of the adjacency matrix. Then, we use the aforementioned random walk method to solve bounds of the greatest eigenvalue of the second-order combinatorial Laplacian matrix. Finally, by comparing the proposed method with existing algorithms in synthetic and real networks, its accuracy and superiority are demonstrated. Full article

For a connected graph G, let $D^L\left(G\right)$ be its distance Laplacian matrix and ${\lambda }_1\left(G\right)\ge {\lambda }_2\left(G\right)\ge \dots \ge {\lambda }_{n-1}\left(G\right)>{\lambda }_n\left(G\right)=0$ be its $D^L$ eigenvalues. The $D^L$ energy of G is defined as $DLE\left(G\right)={\sum }_{i=1}^n\left|{\lambda }_i\left(G\right)-{\displaystyle \frac{2W\left(G\right)}{n}}\right|$, where $W\left(G\right)$ is the Wiener index of G. An important problem in graph energy studies is to determine exact formulations of the energy for specific graph classes and their complements. This paper gives the precise $D^L$ energy formulations of a class of bicyclic graphs $C_2(p,q)$, a class of unicyclic graphs $C_1(p,q)$, and their complements. Moreover, we order the graphs $C_2(p,q)$ on the basic of ${\lambda }_1$, ${\lambda }_{n-1}$, and consider the same problems for their complements. And the ordering of the graphs $C_1(p,q)$ on the basic of ${\lambda }_{n-1}$ and the ordering of their complements on the basics of ${\lambda }_1$, ${\lambda }_{n-1}$ and the $D^L$ energy are obtained. 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

The first Zagreb index of a graph G is defined as the sum of the squares of the degrees of all the vertices in G. The Laplacian spectral radius of a graph G is defined as the largest eigenvalue of the Laplacian matrix of the graph G. In this paper, we first establish inequalities on the first Zagreb index and the Laplacian spectral radius of a graph. Using the ideas of proving the inequalities, we present sufficient conditions involving the first Zagreb index and the Laplacian spectral radius for some Hamiltonian properties of graphs. Full article

Let X be a partitioned matrix and let B its equitable quotient matrix. Consider a simple, undirected, connected graph G of order n. In this paper, we employ a technique based on quotient matrices derived from block-partitioned structures to establish new spectral results for the reciprocal distance signless Laplacian matrix. In particular, we identify a sequence of graphs whose eigenvalues are all integers. Furthermore, we introduce the concept of Harary incidence energy and extend known incidence energy results to the setting of the reciprocal distance signless Laplacian matrix. Finally, we characterize the Harary incidence energy of extremal graphs by examining vertex connectivity through the generalized graph join operation. 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

A new multi-robot path planning (MRPP) algorithm for 2D static environments was developed and evaluated. It combines a roadmap method, utilising the visibility graph (VG), with the algebraic connectivity (second smallest eigenvalue (λ₂)) of the graph’s Laplacian and Dijkstra’s algorithm. The paths depend on the planning order, i.e., they are in sequence path-by-path, based on the measured values of algebraic connectivity of the graph’s Laplacian and the determined weight functions. Algebraic connectivity maintains robust communication between the robots during their navigation while avoiding collisions. The algorithm efficiently balances connectivity maintenance and path length minimisation, thus improving the performance of path finding. It produced solutions with optimal paths, i.e., the shortest and safest route. The devised MRPP algorithm significantly improved path length efficiency across different configurations. The results demonstrated highly efficient and robust solutions for multi-robot systems requiring both optimal path planning and reliable connectivity, making it well-suited in scenarios where communication between robots is necessary. Simulation results demonstrated the performance of the proposed algorithm in balancing the path optimality and network connectivity across multiple static environments with varying complexities. The algorithm is suitable for identifying optimal and complete collision-free paths. The results illustrate the algorithm’s effectiveness, computational efficiency, and adaptability. Full article

This paper provides an efficient method to compute an $LDU$ decomposition of the Laplacian matrix of a connected graph with high relative accuracy. Several applications of this method are presented. In particular, it can be applied to efficiently compute the eigenvalues of the mentioned Laplacian matrix. Moreover, the method can be extended to graphs with weighted edges. Full article

We consider a mathematical model describing the propagation of an epidemic on a geographical network. The size of the outbreak is governed by the initial growth rate of the disease given by the maximal eigenvalue of the epidemic matrix formed by the susceptibles and the graph Laplacian representing the mobility. We use matrix perturbation theory to analyze the epidemic matrix and define a vaccination strategy, assuming vaccination reduces the susceptibles. When mobility and the local disease dynamics have similar time scales, it is most efficient to vaccinate the whole network because the disease grows uniformly. However, if only a few vertices can be vaccinated, then we show that it is most efficient to vaccinate along an eigenvector corresponding to the largest eigenvalue of the Laplacian. We illustrate these results by calculations on a seven-vertex graph and a realistic example of the French rail network. When mobility is slower than the local disease dynamics, the epidemic grows on the vertex with largest number of susceptibles. The epidemic growth rate is more reduced when vaccinating a larger degree vertex; it also depends on the neighboring vertices. This study and its conclusions provide guidelines for the planning of a vaccination campaign on a network at the onset of an epidemic. Full article

In this paper, a distributed algorithm for reaching average consensus is proposed for multi-agent systems with tree communication graph, when the edge weight distribution is unbalanced. First, the problem is introduced as a key topic of core algorithms for several modern scenarios. Then, the relative solution is proposed as a finite-time algorithm, which can be included in any application as a preliminary setup routine, and it is well-suited to be integrated with other adaptive setup routines, thus making the proposed solution useful in several practical applications. A special focus is devoted to the integration of the proposed method with a recent Laplacian eigenvalue allocation algorithm, and the implementation of the overall approach in a wireless sensor network framework. Finally, a worked example is provided, showing the significance of this approach for reaching a more precise average consensus in uncertain scenarios. Full article

In this paper, we study dual quaternion, dual complex unit gain graphs, and their spectral properties in a unified frame of dual unit gain graphs. Unit dual quaternions represent rigid movements in the 3D space, and have wide applications in robotics and computer graphics. Dual complex numbers have found application in brain science recently. We establish the interlacing theorem for dual unit gain graphs, and show that the spectral radius of a dual unit gain graph is always not greater than the spectral radius of the underlying graph, and these two radii are equal if, and only if, the dual gain graph is balanced. By using dual cosine functions, we establish the closed form of the eigenvalues of adjacency and Laplacian matrices of dual complex and quaternion unit gain cycles. We then show the coefficient theorem holds for dual unit gain graphs. Similar results hold for the spectral radius of the Laplacian matrix of the dual unit gain graph too. Full article

In the paper, we introduce the signless Laplacian $ABC$-matrix $\tilde{\mathrm{Q}}(G)=\overline{D}(G)+\tilde{\mathrm{A}}(G)$, where $\overline{D}(G)$ is the diagonal matrix of $ABC$-degrees and $\tilde{\mathrm{A}}(G)$ is the $ABC$-matrix of $G.$ The eigenvalues of the matrix $\tilde{\mathrm{Q}}(G)$ are the signless Laplacian $ABC$-eigenvalues of G. We give some basic properties of the matrix $\tilde{\mathrm{Q}}(G)$, which includes relating independence number and clique number with signless Laplacian $ABC$-eigenvalues. For bipartite graphs, we show that the signless Laplacian $ABC$-spectrum and the Laplacian $ABC$-spectrum are the same. We characterize the graphs with exactly two distinct signless Laplacian $ABC$-eigenvalues. Also, we consider the problem of the characterization of the graphs with exactly three distinct signless Laplacian $ABC$-eigenvalues and solve it for bipartite graphs and, in some cases, for non-bipartite graphs. We also introduce the concept of the trace norm of the matrix $\tilde{\mathrm{Q}}(G)-{\displaystyle \frac{tr(\tilde{\mathrm{Q}}(G))}{n}}I$, called the signless Laplacian $ABC$-energy of G. We obtain some upper and lower bounds for signless Laplacian $ABC$-energy and characterize the extremal graphs attaining it. Further, for graphs of order at most 6, we compare the signless Laplacian energy and the $ABC$-energy with the signless Laplacian $ABC$-energy and found that the latter behaves well, as there is a single pair of graphs with the same signless Laplacian $ABC$-energy unlike the 26 pairs of graphs with same signless Laplacian energy and eight pairs of graphs with the same $ABC$-energy. Full article