Algebraic Connectivity Research Articles

Analyzing the Connection of Linear Algebra: Enhancing Visual Reasoning through Vector Spaces

Analyzing the Connection of Linear Algebra: Enhancing Visual Reasoning through Vector Spaces

Conceptualizing algebraic connections: Honoring voices from future mathematics teachers and the instructors who teach them

Conceptualizing algebraic connections: Honoring voices from future mathematics teachers and the instructors who teach them

Extreme values of the Fiedler vector on trees

Let G be a tree on n vertices and let L=D−A denote the Laplacian matrix on G. The second-smallest eigenvalue λ2(G)>0, also known as the algebraic connectivity, as well as the associated eigenvector have been of substantial interest. We investigate the question of when the maxima and minima of an associated eigenvector are assumed at the endpoints of the longest path in G. Our results also apply to more general graphs that ‘behave globally’ like a tree but can exhibit more complicated local structure. The crucial new ingredient is a reproducing formula for eigenvectors of graphs.

Spectral Bipartition via Gap Cut on DNA Sequences

Deoxyribonucleic Acid (DNA) and graph partitioning are two distinct fields of study which can be linked in the structure of biological networks. Graph partitioning has been extensively studied but not its application in the biological field. This research explored on the application of spectral graph partitioning in DNA splicing, aiming to simulate the cleavage of DNA by performing spectral bipartition on DNA sequences in a DNA splicing system. This research incorporates Fiedler theory and algebraic graph theory, which are commonly utilized in network analysis and the analysis of graph connectivity. Some DNA sequences of even length are selected and expressed in graphical representations. The adjacency matrix, Laplacian matrix, and degree matrix are computed from the graphs, as well as the Fiedler value and Fiedler vector associated with the graphs. Gap cut is used as a method of spectral bipartition which produces two partitions of DNA sequence of unequal lengths. The generalizations of gap cut on DNA sequences of even length are provided as lemmas and theorem.

Graphs of Degree at Least \({3}\) with Minimum Algebraic Connectivity

Graphs of Degree at Least \({3}\) with Minimum Algebraic Connectivity

Effects on the algebraic connectivity of weighted graphs under edge rotations

For a weighted graph G, the rotation of an edge uv1 from v1 to a vertex v2 is defined as follows: delete the edge uv1, set w(uv2) as w(uv1)+w(uv2) if uv2 is an edge of G; otherwise, add a new edge uv2 and set w(uv2)=w(uv1), where w(uv1) and w(uv2) are the weights of the edges uv1 and uv2, respectively. In this paper, effects on the algebraic connectivity of weighted graphs under edge rotations are studied. For a weighted graph, a sufficient condition for an edge rotation to reduce its algebraic connectivity and a necessary condition for an edge rotation to improve its algebraic connectivity are proposed based on Fiedler vectors of the graph. As applications, we show that, by using a series of edge rotations, a pair of pendent paths (a pendent tree) of a weighted graph can be transformed into one pendent path (pendent edges attached at a common vertex) of the graph with the algebraic connectivity decreasing (increasing) monotonically. These results extend previous findings of reducing the algebraic connectivity of unweighted graphs by using edge rotations.

Learners’ algebraic and geometric connections when solving Euclidean geometry riders

In this article, we explored Grade 11 learners’ algebraic and geometric connections when solving Euclidean geometry riders. A qualitative interpretive case study design was followed. Thirty Grade 11 learners from a non-fee-paying secondary school in the Capricorn North district of South Africa were conveniently sampled to participate in this study. Data were collected through learners’ responses to classwork, homework exercises, and task-based interviews. Data were analysed thematically. The findings revealed that to solve Euclidean geometry riders successfully, learners need to establish the feature connections embedded in the given figure or diagram. The ability to make feature connections provides a point of departure in the solution process of a geometric problem.Contribution: Once the feature connection is established, other connections will subsequently emerge. In addition, the reversibility connections become a form of feature connection when solving Euclidean geometry riders. Therefore, we recommend that mathematics teachers emphasise and use mathematical connections in their daily teaching of mathematics.

Node Centrality Based on Edge Dynamics in a Chaotic Network

This paper introduces a novel node centrality index based on edge dynamics in a complex network: adjacent edge index. Through the analysis of network dynamics and topology, we provide a neat index to qualify node importance, which is calculated by the eigenvector components corresponding to the Fiedler value of the network. Notably, it identifies nodes whose adjacent edge sets play a significant role in network synchronization. The index indicates that not only a node’s degree but also its location should be considered. To demonstrate the effectiveness of the proposed index, consider a double-star network and a Jazz Musicians network in which chaotic Chua’s circuits as nodes are coupled. By factoring in the dynamic information exchange between nodes and their neighbors, this index offers valuable insights for optimizing strategies in engineering areas such as efficient information dissemination.

Dynamics of a susceptible-infected-recovered model on complex networks with interregional migration.

We present a susceptible-infected-recovered model based on a dynamic flow network that describes the epidemic process on complex metapopulation networks. This model views population regions as interconnected nodes and describes the evolution of each region using a system of differential equations. The next-generation matrix method is used to derive the global basic reproduction number for three cases: a general network with homogeneous infection rates in all regions, a fully connected network, and a star network with heterogeneous infection and recovery rates. For the homogeneous case, we show that this global basic reproduction number is independent of the migration rates between regions. However, the rate of convergence of each region to an equilibrium state exhibits a much larger variance in random (Erdős-Rényi) networks compared to small-scale (Barabási-Albert) networks. For the general heterogeneous case, we report interesting results, namely that the global basic reproduction number decays exponentially with respect to the smallest nonzero Laplacian eigenvalue (algebraic connectivity). Furthermore, we demonstrate both analytically and numerically that as the network's algebraic connectivity increases, either by increasing the average node degree of each region or the global migration rate, the global basic reproduction number decreases and converges to the ratio of the average local infection rate to the average local recovery rate, meaning that the lower bound of the global basic reproduction rate does not equal the mean of local basic reproduction rates.

Garland's method for token graphs

The k-th token graph of a graph G=(V,E) is the graph Fk(G) whose vertices are the k-subsets of V and whose edges are all pairs of k-subsets A,B such that the symmetric difference of A and B forms an edge in G. Let L(G) be the Laplacian matrix of G, and Lk(G) be the Laplacian matrix of Fk(G). It was shown by Dalfó, Duque, Fabila-Monroy, Fiol, Huemer, Trujillo-Negrete, and Zaragoza Martínez that for any graph G on n vertices and any 0≤ℓ≤k≤⌊n/2⌋, the spectrum of Lℓ(G) is contained in that of Lk(G).Here, we continue to study the relation between the spectrum of Lk(G) and that of Lk−1(G). In particular, we show that, for 1≤k≤⌊n/2⌋, any eigenvalue λ of Lk(G) that is not contained in the spectrum of Lk−1(G) satisfiesk(λ2(L(G))−k+1)≤λ≤kλn(L(G)), where λ2(L(G)) is the second smallest eigenvalue of L(G) (also known as the algebraic connectivity of G), and λn(L(G)) is its largest eigenvalue. Our proof relies on an adaptation of Garland's method, originally developed for the study of high-dimensional Laplacians of simplicial complexes.

Algebraic Connectivity of Power Graphs of Finite Cyclic Groups

Algebraic Connectivity of Power Graphs of Finite Cyclic Groups

SE(3) Synchronization by eigenvectors of dual quaternion matrices

Abstract In synchronization problems, the goal is to estimate elements of a group from noisy measurements of their ratios. A popular estimation method for synchronization is the spectral method. It extracts the group elements from eigenvectors of a block matrix formed from the measurements. The eigenvectors must be projected, or ‘rounded’, onto the group. The rounding procedures are constructed ad hoc and increasingly so when applied to synchronization problems over non-compact groups. In this paper, we develop a spectral approach to synchronization over the non-compact group $\mathrm{SE}(3)$, the group of rigid motions of $\mathbb{R}^{3}$. We based our method on embedding $\mathrm{SE}(3)$ into the algebra of dual quaternions, which has deep algebraic connections with the group $\mathrm{SE}(3)$. These connections suggest a natural rounding procedure considerably more straightforward than the current state of the art for spectral $\mathrm{SE}(3)$ synchronization, which uses a matrix embedding of $\mathrm{SE}(3)$. We show by numerical experiments that our approach yields comparable results with the current state of the art in $\mathrm{SE}(3)$ synchronization via the spectral method. Thus, our approach reaps the benefits of the dual quaternion embedding of $\mathrm{SE}(3)$ while yielding estimators of similar quality.

Attainable bounds for algebraic connectivity and maximally connected regular graphs

AbstractWe derive attainable upper bounds on the algebraic connectivity (spectral gap) of a regular graph in terms of its diameter and girth. This bound agrees with the well‐known Alon–Boppana–Friedman bound for graphs of even diameter, but is an improvement for graphs of odd diameter. For the girth bound, we show that only Moore graphs can attain it, and these only exist for well‐known special cases. For the diameter bound, we use a combination of stochastic algorithms and exhaustive search to find graphs which attain it. For 3‐regular graphs, we find attainable graphs for all diameters up to and including (the case of is open). These graphs are extremely rare and also have high girth; for example, we found exactly 45 distinct cubic graphs on 44 vertices attaining the upper bound when ; all have girth 8. We also exhibit several infinite families attaining the upper bound with respect to diameter or girth. In particular, when is a power of prime, we construct a ‐regular graph having diameter 4 and girth 6 which attains the upper bound with respect to diameter.

Spectral reordering for faster elasticity simulations

We present a novel method for faster physics simulations of elastic solids. Our key idea is to reorder the unknown variables according to the Fiedler vector (i.e., the second-smallest eigenvector) of the combinatorial Laplacian. It is well known in the geometry processing community that the Fiedler vector brings together vertices that are geometrically nearby, causing fewer cache misses when computing differential operators. However, to the best of our knowledge, this idea has not been exploited to accelerate simulations of elastic solids which require an expensive linear (or non-linear) system solve at every time step. The cost of computing the Fiedler vector is negligible, thanks to an algebraic Multigrid-preconditioned Conjugate Gradients (AMGPCG) solver. We observe that our AMGPCG solver requires approximately 1 s for computing the Fiedler vector for a mesh with approximately 50K vertices or 100K tetrahedra. Our method provides a speed-up between 10%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10\\%$$\\end{document} – 30%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$30\\%$$\\end{document} at every time step, which can lead to considerable savings, noting that even modest simulations of elastic solids require at least 240 time steps. Our method is easy to implement and can be used as a plugin for speeding up existing physics simulators for elastic solids, as we demonstrate through our experiments using the Vega library and the ADMM solver, which use different algorithms for elasticity.

Innovative graph-based video processing methodology for collapse early warning of historic masonry buildings

In the present study an innovative video processing methodology is proposed for application to vibration data of historic masonry structures. The proposed methodology is based on graph theory and topology analysis applied to magnified videos in search of effective parameters for early-warning signals before the collapse of structures in case of earthquakes. The proposed method was validated through seismic tests of a brick-masonry mockup representing a vault of the mosque in the Palace of the Dey, Algiers. In particular, the seismic tests were carried out at increasing earthquake intensity up to the final collapse of the mockup. After processing the videos of the seismic tests by motion magnification method, the magnified video frames were transformed into a graph of the structure. Finally, several graph indices were calculated and monitored during the vibration. The monitored parameters were analyzed in search of potential threshold values suitable to generate an early warning signal. In particular, the inverse algebraic connectivity provided an early warning signal in the order of a few seconds before collapse. This was validated by comparison with an analogous signal provided at similar time by an accurate displacement lab measurements system based on optical markers positioned at several points of the tested mock-up.

On the harmonic index and algebraic connectivity

On the harmonic index and algebraic connectivity

On the algebraic connectivity of some token graphs

The k-token graph Fk(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_k(G)$$\\end{document} of a graph G is the graph whose vertices are the k-subsets of vertices from G, two of which are adjacent whenever their symmetric difference is a pair of adjacent vertices in G. It was proved that the algebraic connectivity of Fk(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_k(G)$$\\end{document} equals the algebraic connectivity of G with a proof using random walks and interchange of processes on a weighted graph. However, no algebraic or combinatorial proof is known, and it would be a hit in the area. In this paper, we algebraically prove that the algebraic connectivity of Fk(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_k(G)$$\\end{document} equals the one of G for new infinite families of graphs, such as trees, some graphs with hanging trees, and graphs with minimum degree large enough. Some examples of these families are the following: the cocktail party graph, the complement graph of a cycle, and the complete multipartite graph.

Fiedler value: The cumulated dynamical contribution value of all edges in a complex network.

Fiedler value, as the minimal real part of (or the minimal) nonzero Laplacian eigenvalue, garners significant attention as a metric for evaluating network topology and its dynamics. In this paper, we address the quantification relation between Fiedler value and each edge in a directed complex network, considering undirected networks as a special case. We propose an approach to measure the dynamical contribution value of each edge. Interestingly, these contribution values can be both positive and negative, which are determined by the left and right Fiedler vectors. Further, we show that the cumulated dynamical contribution value of all edges is exactly the Fiedler value. This provides a promising angle on the Fiedler value in terms of dynamics and network structure. Therefore, the percentage of contribution of each edge to the Fiedler value is quantified. Numerical results reveal that network dynamics is significantly influenced by a small fraction of edges, say, one single directed edge contributes to over 90% of the Fiedler value in the Cat Cerebral Cortex network.

Robust H∞ consensus for uncertain linear multi-agent systems on directed topologies: A dynamic event-triggered strategy

Considering the asymmetry of the system matrix caused by directed communication networks, a dynamic event-triggered control protocol for the robust [Formula: see text] consensus of uncertain linear multi-agent systems accompanied by external disturbances on directed topologies is proposed. Through the introduction of the controlled output and generalized algebraic connectivity for directed graphs, the consensus problem is converted to the event-triggered stabilization of the uncertain linear closed-loop system with disturbance. On the basis of the relative state of neighbor agents at event-times and the introduced dynamic variables, a dynamic event-triggered control protocol is put forward, and the linear matrix inequality sufficient conditions for the system to achieve robust [Formula: see text] consensus are obtained after a rigorous proof. Then, linear matrix inequalities are further decoupled to greatly reduce the computational complexity, and the parameters of the control protocol are determined by solving the decoupled linear matrix inequalities. Finally, simulations are applied for the verification of conclusions.

The Chvátal–Gomory procedure for integer SDPs with applications in combinatorial optimization

AbstractIn this paper we study the well-known Chvátal–Gomory (CG) procedure for the class of integer semidefinite programs (ISDPs). We prove several results regarding the hierarchy of relaxations obtained by iterating this procedure. We also study different formulations of the elementary closure of spectrahedra. A polyhedral description of the elementary closure for a specific type of spectrahedra is derived by exploiting total dual integrality for SDPs. Moreover, we show how to exploit (strengthened) CG cuts in a branch-and-cut framework for ISDPs. Different from existing algorithms in the literature, the separation routine in our approach exploits both the semidefinite and the integrality constraints. We provide separation routines for several common classes of binary SDPs resulting from combinatorial optimization problems. In the second part of the paper we present a comprehensive application of our approach to the quadratic traveling salesman problem (QTSP). Based on the algebraic connectivity of the directed Hamiltonian cycle, two ISDPs that model the QTSP are introduced. We show that the CG cuts resulting from these formulations contain several well-known families of cutting planes. Numerical results illustrate the practical strength of the CG cuts in our branch-and-cut algorithm, which outperforms alternative ISDP solvers and is able to solve large QTSP instances to optimality.