Search Results (143)

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

Multi-agent systems (MAS) typically model interaction topologies using directed or undirected graphs when analyzing consensus convergence rates. However, as system complexity increases, purely directed or undirected networks may be insufficient to capture interaction heterogeneity. This paper adopts hybrid networks as interaction topology to investigate strategies for improving consensus convergence rates. We propose the Hermitian Kirchhoff index, a novel metric based on resistance distance, to quantify the consensus convergence rates and establish its theoretical justification. We then examine how adding or removing edges/arcs affects the Hermitian Kirchhoff index, employing first-order eigenvalue perturbation analysis to relate these changes to algebraic connectivity and its associated eigenvectors. Numerical simulations corroborate the theoretical findings and demonstrate the effectiveness of the proposed approach. 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

The growing importance of social networks has led to increased research into trust estimation and interpretation among network entities. It is important to predict the trust score between users in order to minimize the risks in user interactions. This article enables the identification of the most reliable and least reliable entities in a network by expressing trust scores numerically. In this paper, the social network is modeled as a graph, and trust scores are calculated by taking the powers of the ratio matrix between entities and summing them. Taking the power of the proportion matrix based on the number of entities in the network requires a lot of arithmetic load. After taking the powers of the eigenvalues of the ratio matrix, these are multiplied by the eigenvector matrix to obtain the power of the ratio matrix. In this way, the arithmetic cost required for calculating trust between entities is reduced. This paper calculates the trust score between entities using linear algebra techniques to reduce the arithmetic load. Trust detection algorithms use shortest paths and similar methods to eliminate paths that are deemed unimportant, which makes the result questionable because of the loss of data. The novelty of this method is that it calculates the trust score without the need for explicit path numbering and without any data loss. 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 second-order linear differential operator A is defined on a tree of arbitrary topology. Any internal vertex P of the tree divides the tree into $m_p$ branches. The restrictions $A_i,i=1,\dots ,m_p$ of the operator A on each of these branches, where the vertex P is considered the root of the branch and the Dirichlet boundary condition is specified at the root. These restrictions must be, in a sense, asymmetric (not similar) to each other. Thus, the distinguished class of differential operators A turns out to have only simple eigenvalues. Moreover, the matching conditions at the internal vertices of the graph contain a set of parameters. These parameters are interpreted as stiffness coefficients. This paper proves that a finite set of eigenvalues allows one to uniquely restore the set of stiffness coefficients. The novelty of the work is the fact that it is sufficient to know a finite set of eigenvalues of intermediate Weinstein problems for uniquely restoring the required stiffness coefficients. We not only describe the results of selected studies but also compare them with each other and establish interconnections. Full article

We address the problem of the distributed computation of arbitrary functions of two correlated sources, $X_1$ and $X_2$, residing in two distributed source nodes, respectively. We exploit the structure of a computation task by coding source characteristic graphs (and multiple instances using the n-fold OR product of this graph with itself). For regular graphs and general graphs, we establish bounds on the optimal rate—characterized by the chromatic entropy for the n-fold graph products—that allows a receiver for asymptotically lossless computation of arbitrary functions over finite fields. For the special class of cycle graphs (i.e., 2-regular graphs), we establish an exact characterization of chromatic numbers and derive bounds on the required rates. Next, focusing on the more general class of d-regular graphs, we establish connections between d-regular graphs and expansion rates for n-fold graph products using graph spectra. Finally, for general graphs, we leverage the Gershgorin Circle Theorem (GCT) to provide a characterization of the spectra, which allows us to derive new bounds on the optimal rate. Our codes leverage the spectra of the computation and provide a graph expansion-based characterization to succinctly capture the computation structure, providing new insights into the problem of distributed computation of arbitrary functions. Full article

The entanglement entropy of a random tensor network (RTN) is studied using tools from free probability theory. Random tensor networks are simple toy models that help in understanding the entanglement behavior of a boundary region in the anti-de Sitter/conformal field theory (AdS/CFT) context. These can be regarded as specific probabilistic models for tensors with particular geometry dictated by a graph (or network) structure. First, we introduce a model of RTN obtained by contracting maximally entangled states (corresponding to the edges of the graph) on the tensor product of Gaussian tensors (corresponding to the vertices of the graph). The entanglement spectrum of the resulting random state is analyzed along a given bipartition of the local Hilbert spaces. The limiting eigenvalue distribution of the reduced density operator of the RTN state is provided in the limit of large local dimension. This limiting value is described through a maximum flow optimization problem in a new graph corresponding to the geometry of the RTN and the given bipartition. In the case of series-parallel graphs, an explicit formula for the limiting eigenvalue distribution is provided using classical and free multiplicative convolutions. The physical implications of these results are discussed, allowing the analysis to move beyond the semiclassical regime without any cut assumption, specifically in terms of finite corrections to the average entanglement entropy of the RTN. Full article

While the adjacency tensor of a bipartite 2-graph is a nonnegative biquadratic tensor, it is inherently reducible. To address this limitation, we introduce the concept of quasi-irreducibility in this paper. The adjacency tensor of a bipartite 2-graph is quasi-irreducible if that bipartite 2-graph is not bi-separable. This new concept reveals important spectral properties: although all M⁺-eigenvalues are M⁺⁺-eigenvalues for irreducible nonnegative biquadratic tensors, the M⁺-eigenvalues of a quasi-irreducible nonnegative biquadratic tensor can be either M⁰-eigenvalues or M⁺⁺-eigenvalues. Furthermore, we establish a max-min theorem for the M-spectral radius of a nonnegative biquadratic tensor. Full article

This paper studies the problem of time-varying formation-tracking control for a class of nonlinear multi-agent systems. A distributed adaptive controller that avoids the global non-zero minimum eigenvalue is designed for heterogeneous systems in which leaders and followers contain different nonlinear terms, and which relies only on the relative errors between adjacent agents. By adopting the Riccati inequality method, the adaptive adjustment factor in the controller is designed to solve the problem of automatically adjusting relative errors based solely on local information. Unlike existing research on time-varying formations with fixed and switching topologies, the method of jointly connected topological graphs is adopted to enable nonlinear followers to track the trajectories of leaders with different nonlinear terms and simultaneously achieve the control objective of the desired time-varying formation. The stability of the system under the jointly connected graph is proved by the Lyapunov stability proof method. Finally, numerical simulation experiments confirm the effectiveness of the proposed control method. Full article

The spectrum of a graph $\mathrm{\Gamma }$, denoted by $Spec\left(\mathrm{\Gamma }\right)$, is the multiset of eigenvalues of its adjacency matrix. A Cayley graph $Cay(G,S)$ of a finite group G is called Cay-DS (Cayley graph determined by its spectrum) if, for any other Cayley graph $Cay(G,T)$, $Spec\left(Cay\right(G,S\left)\right)=Spec\left(Cay\right(G,T\left)\right)$ implies $Cay(G,S)\cong Cay(G,T)$. A group G is said to be Cay-DS if all Cayley graphs of G are Cay-DS. An interesting open problem in the area of algebraic graph theory involves characterizing finite Cay-DS groups or constructing non-isomorphic Cayley graphs of a non-Cay-DS group that share the same spectrum. The present paper contributes to parts of this problem of metacyclic groups $M_{8p}$ of order $8p$ (with center of order 4), where p is an odd prime, in terms of irreducible characters, which are first presented. Then some new families of pairwise non-isomorphic Cayley graph pairs of $M_{8p}$ ($p\ge 5$) with the same spectrum are found. As a conclusion, this paper concludes that $M_{8p}$ is Cay-DS if and only if $p=3$. Full article

This paper explores the common neighborhood energy ($E_{CN}\left(\mathsf{\Gamma }\right)$) of graphs derived from the dihedral group $D_{2n}$ and generalized quaternion group $Q_{4n}$, specifically the non-commuting graph (NCM-graph) and the commuting graph (CM-graph). Studying graphs associated with groups offers a powerful approach to translating algebraic properties into combinatorial structures, enabling the application of graph-theoretic tools to understand group behavior. The common neighborhood energy, defined as the sum of the absolute values of the eigenvalues of the common neighborhood (CN) matrix, i.e., ${\sum }_{i=1}^p\left|{\zeta }_i\right|$, where ${\left\{{\zeta }_i\right\}}_{i=1}^p$ are the CN eigenvalues, provides insights into the structural properties of these graphs. We derive explicit formulas for the CN characteristic polynomials and corresponding CN eigenvalues for both the NCM-graph and CM-graph as functions of n. Consequently, we establish closed-form expressions for the $E_{CN}$ of these graphs, which are parameterized by n. The validity of our theoretical results is confirmed through computational examples. This study contributes to the spectral analysis of algebraic graphs, demonstrating a direct connection between the group-theoretic structure of $D_{2n}$ and $Q_{4n}$, as well as the combinatorial energy of their associated graphs, thus furthering the understanding of group properties through spectral graph theory. Full article

Let G be a graph with n vertices, and let $d_G\left(u\right)$ denote the degree of vertex u in $G.$ The maximum degree matrix ${\mathcal{M}}_G$ of G is the square matrix of order n whose $(u,v)$-entry is equal to $max{d}_G\left(u\right),d_G\left(v\right)$ if vertices u and v are adjacent in G, and zero otherwise. Let $B_{p,q,r}$ be the graph obtained from the complete graph $K_p$ by removing an edge $uv$, and identifying vertices u and v with the end vertices $u\prime$ and $v\prime$ of the paths $P_q$ and $P_r$, respectively. Let $\mathbb{G}n,d$ denote the set of simple, connected graphs with n vertices and diameter d. A graph in $\mathbb{G}n,d$ that attains the largest spectral radius of the maximum degree matrix is called a maximizing graph. In this paper, we first characterize the spectrum of the maximum degree matrix for graphs of the form $B_{n-i+2,i,d-i}$, where $1\le i\le \lfloor \frac{d}{2}\rfloor$. Furthermore, for $d\ge 2,$ we prove that the maximizing graph in $\mathbb{G}n,d$ is $Bn-d+2,\lfloor \frac{d}{2}\rfloor ,\lceil \frac{d}{2}\rceil .$ Finally, if $d\ge 4$ is an even integer, then the spectral radius of the maximum degree matrix in $B_{n-d+2,\lfloor \frac{d}{2}\rfloor ,\lceil \frac{d}{2}\rceil }$ can be computed as the largest eigenvalue of a symmetric tridiagonal matrix of order ${\displaystyle \frac{d}{2}}+1$. 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