spectral radius and signless laplacian spectral radius of strongly connected digraphs abstract let d be a strongly connected digraph and a d be the adjacency matrix of d let diag d be the diagonal matrix with outdegrees of the vertices of d and q d diag d a d be the signless laplacian matrix of d the spectral radius of q d is called the signless laplacian spectral radius of d denoted by q d in this paper we give a sharp bound on q d where d has a given outdegree sequence and compare the bound with known bounds we establish some sharp upper or lower bound on q d with some given parameter such as clique number girth or vertex connectivity and characterize the extremal graph in addition we also determine the unique digraph which achieves the minimum or maximum the second minimum or maximum the third minimum the fourth minimum spectral radius and signless laplacian spectral radius among all strongly connected digraphs and answer the open problem proposed by lin and shu 14

distributed formation control of multi agent systems using complex laplacian the paper concentrates on the fundamental coordination problem that requires a network of agents to achieve a specific but arbitrary formation shape a new technique based on complex laplacian is introduced to address the problems of which formation shapes specified by inter agent relative positions can be formed and how they can be achieved with distributed control ensuring global stability concerning the first question we show that all similar formations subject to only shape constraints are those that lie in the null space of a complex laplacian satisfying certain rank condition and that a formation shape can be realized almost surely if and only if the graph modeling the inter agent specification of the formation shape is 2 rooted concerning the second question a distributed and linear control law is developed based on the complex laplacian specifying the target formation shape and provable existence conditions of stabilizing gains to assign the eigenvalues of the closed loop system at desired locations are given moreover we show how the formation shape control law is extended to achieve a rigid formation if a subset of knowledgable agents knowing the desired formation size scales the formation while the rest agents do not need to re design and change their control laws

laplacian graph eigenvectors abstract if g is a graph its laplacian is the difference of the diagonal matrix of its vertex degrees and its adjacency matrix the main thrust of the present article is to prove several laplacian eigenvector principles which in certain cases can be used to deduce the effect on the spectrum of contracting adding or deleting edges and or of coalescing vertices one application is the construction of two isospectral graphs on 11 vertices having different degree sequences only one of which is bipartite and only one of which is decomposable

signless laplacians of finite graphs we survey properties of spectra of signless laplacians of graphs and discuss possibilities for developing a spectral theory of graphs based on this matrix for regular graphs the whole existing theory of spectra of the adjacency matrix and of the laplacian matrix transfers directly to the signless laplacian and so we consider arbitrary graphs with special emphasis on the non regular case the results which we survey old and new are of two types a results obtained by applying to the signless laplacian the same reasoning as for corresponding results concerning the adjacency matrix b results obtained indirectly via line graphs among other things we present eigenvalue bounds for several graph invariants an interpretation of the coefficients of the characteristic polynomial a theorem on powers of the signless laplacian and some remarks on star complements

decentralized control of vehicle formations this paper investigates a method for decentralized stabilization of vehicle formations using techniques from algebraic graph theory the vehicles exchange information according to a pre specified communication digraph g a feedback control is designed using relative information between a vehicle and its in neighbors in g we prove that a necessary and sufficient condition for an appropriate decentralized linear stabilizing feedback to exist is that g has a rooted directed spanning tree we show the direct relationship between the rate of convergence to formation and the eigenvalues of the directed laplacian of g various special situations are discussed including symmetric communication graphs and formations with leaders several numerical simulations are used to illustrate the results

leader follower formation via complex laplacian the paper introduces complex valued laplacians for graphs whose edges are attributed with complex weights and studies the leader follower formation problem based on complex laplacians the main goal is to control the shape of a planar formation of point agents in the plane using simple and linear interaction rules related to complex laplacians we present a characterization of complex laplacians that preserve a specific planar formation as an equilibrium solution for both single integrator kinematics and double integrator dynamics planar formations under study are subject to translation rotation and scaling in the plane but can be determined by two co leaders in leader follower networks furthermore when a complex laplacian does not result in an asymptotically stable behavior of the multi agent system we show that a stabilizing matrix which updates the complex weights exists to asymptotically stabilize the system while preserving the equilibrium formation also algorithms are provided to find stabilizing matrices finally simulations are presented to illustrate our results

sharp bounds for the signless laplacian spectral radius in terms of clique number abstract in this paper we present a sharp upper and lower bounds for the signless laplacian spectral radius of graphs in terms of clique number moreover the extremal graphs which attain the upper and lower bounds are characterized in addition these results disprove the two conjectures on the signless laplacian spectral radius in 8

unified cooperative control of multiple agents on a sphere for different spherical patterns in this technical note we consider the cooperative control of multiple agents on a sphere provide the appropriate attraction repulsion interactions between agents with introduction of the state dependent repulsive coefficients and design the unified control laws for agents to achieve three fundamental spherical configurations or patterns rendezvous uniform deployment and formation using both the first and second order models we present the analysis of the stability scaling and geometric patterns of the models on the sphere under different attraction repulsion parameters with illustration of the geometrical patterns and the geometrical isomerism by simulations the closed loop dynamics of agents are invariant with respect to group rotation actions on the sphere

bounds for the spectral radius of nonnegative matrices abstract we give sharp upper and lower bounds for the spectral radius of a nonnegative matrix with all row sums positive using its average 2 row sums and characterize the equality cases if the matrix is irreducible we compare these bounds with the known bounds using the row sums by examples we also apply these bounds to various matrices associated with a graph including the adjacency matrix the signless laplacian matrix and some distance based matrices some known results are generalized and improved

a conjecture on the diameter and signless laplacian index of graphs abstract a bug bug p q 1 q 2 is a graph obtained from a complete graph k p by deleting an edge uv and attaching paths p q 1 and p q 2 at u and v respectively in this paper we show that for connected graphs g of order n with signless laplacian index q 1 g and diameter diam g q 1 g diam g is maximized for and only for the graph bug n 2 2 p q where p d 2 q d 2 and d n 1 2 this solves a conjecture in 6 on the signless laplacian index involving the diameter

hybrid control for connectivity preserving flocking in this technical note we address the combined problem of motion and network topology control in a group of mobile agents with common objective the flocking behavior of the group instead of assuming network connectivity we enforce it by means of distributed topology control that decides on both deletion and creation of communication links between agents adapting the network to the group s spatial distribution with this protocol ensuring network connectivity a decentralized motion controller aligns agent velocity vectors and regulates inter agent distances to maintain existing network links the stability of the flocking controller is established in continuous time by means of an observability argument on a quadratic form of the graph laplacian that exploits the time delay between link deletion and creation caused by the topology control protocol which induces a dwell time between network switches

on the signless laplacian index and radius of graphs abstract a bag b a g p q is a graph obtained from a complete graph k p by replacing an edge uv by a path p q in this paper we show that for all the connected graphs of order n 5 with signless laplacian index q 1 g and radius r a d g q 1 g r a d g is maximum for and only for the graph b a g n 2 s 3 2 s 1 where s n 4 this solves a conjecture in 6 on the signless laplacian index involving the radius

bounds and conjectures for the signless laplacian index of graphs abstract using the autographix system we obtain conjectures of the form l n q 1 i g u n where q 1 denotes the signless laplacian index of graph g is one the four operations i g is another invariant chosen among minimum average and maximum degree average distance diameter radius girth proximity remoteness vertex edge and algebraic connectivities independence number domination number clique number chromatic number and matching number randic index l n and u n are best possible lower and upper bounds function of the order n of g algebraic conjectures are obtained in 120 cases out of 152 and structural conjectures in 12 of the remaining cases these conjectures are known immediate or proved in this paper except for 17 of them which remain open

on the rendezvous problem for multiple nonholonomic agents in this note a decentralized feedback control strategy that drives a system of multiple nonholonomic unicycles to a rendezvous point in terms of both position and orientation is introduced the proposed nonholonomic control law is discontinuous and time invariant and using tools from nonsmooth lyapunov theory and graph theory the stability of the overall system is examined similarly to the linear case the convergence of the multi agent system relies on the connectivity of the communication graph that represents the inter agent communication topology the control law is first defined in order to guarantee connectivity maintenance for an initially connected communication graph moreover the cases of static and dynamic communication topologies are treated as corollaries of the proposed framework

the signless laplacian spectral radius of tricyclic graphs and trees with k pendant vertices abstract in this paper we consider the following problem of all tricyclic graphs or trees of order n with k pendant vertices n k fixed which achieves the maximal signless laplacian spectral radius we determine the graph with the largest signless laplacian spectral radius among all tricyclic graphs with n vertices and k pendant vertices then we show that the maximal signless laplacian spectral radius among all trees of order n with k pendant vertices is obtained uniquely at t n k where t n k is a tree obtained from a star k 1 k and k paths of almost equal lengths by joining each pendant vertex to one end vertex of one path we also discuss the signless laplacian spectral radius of t n k and give some results

asymptotic incidence energy of lattices abstract the energy of a graph g arising in chemical physics denoted by e g is defined as the sum of the absolute values of the eigenvalues of g as an analogue to e g the incidence energy ie g defined as the sum of the singular values of the incidence matrix of g is a much studied quantity with well known applications in chemical physics in this paper based on the results by yan and zhang 2009 we propose the incidence energy per vertex problem for lattice systems and present the closed form formulae expressing the incidence energy of the hexagonal lattice triangular lattice and 3 3 4 2 lattice respectively moreover we show that the incidence energy per vertex of lattices is independent of the toroidal cylindrical and free boundary conditions in particular the explicit asymptotic values of the incidence energy in these lattices are obtained by utilizing the applications of analysis approach with the help of calculational software

combined effects of concave and convex nonlinearities in some elliptic problems abstract this paper deals with a class of semilinear elliptic dirichlet boundary value problems where the combined effects of a sublinear and a superlinear term allow us to establish some existence and multiplicity results

brief paper a novel approach to coordination of multiple robots with communication failures via proximity graph in this paper we investigate the coordination problem of multiple robots with limited communication ranges and communication failures a novel rendezvous algorithm via proximity graph is developed so that the robot group can achieve rendezvous when the communication links satisfy an ergodic assumption the convergence proof of the algorithm is established based on the tools from rooted graph theory the effectiveness of the algorithm is illustrated by numerical examples in a 3d space

on ordinary and signless laplacian spectral radius of graphs with fixed number of branch vertices the unique graphs with maximum spectral radius and signless laplacian spectral radius are determined among trees unicyclic graphs and bicyclic graphs respectively with fixed numbers of vertices and branch vertices vertices of degree at least 3 using a unified approach

styled velocity flocking of autonomous vehicles a systematic design this paper develops a measure theoretic approach for generating the collective behavior of flocking of planar autonomous vehicles under all to all communication scheme to obtain flocking protocol we formulate the notion of state of styled velocity flocking for the target state of a design problem instead of considering solution ansatzes this notion captures a broader class of flocking behaviors of non parallel motions as well as embodies the two important performances of cohesion maintenance and collision avoidance to pose a control design problem a measure valued transition equation is obtained to induce an auxiliary one body continuum model from the exact multi body particle model of the collective system the design problem is then formulated on the auxiliary continuum model and a systematic design procedure is presented the state of styled velocity flocking of the particle model under the obtained flocking protocol is then verified with an important result on the continuous dependence on initial data of solutions of the measured valued transition equation application to elliptical flocking of mobile robots is presented to illustrate the novelty of the presented theory

cooperative output regulation of heterogeneous multi agent systems an h_ infty criterion in this note the cooperative output regulation problem of heterogeneous linear multi agent systems is investigated our approach is based on internal model and dynamic error feedback among the agents under some standard assumptions the problem is solved by an internal model and a stabilizing h controller we showed that the exo signal free system is stabilized if each controlled agent is robust in the sense of h norm to resist the disturbance measured by the second largest eigenvalue of the graph adjacency matrix then the synthesis problem is solved using algebraic riccati equation furthermore we showed that this condition is a generalization of existing results

the constrained rayleigh quotient with a general orthogonality constraint and an eigen balanced laplacian matrix the greatest lower bound and applications in cooperative control problems mathematically the rayleigh quotient is defined as the quadratic function of a symmetric matrix and a nonzero usually unconstrained variable vector in this paper we consider the constrained rayleigh quotient in which the variable vector has the orthogonality constraint i e it is constrained to be orthogonal to a nonzero vector this nonzero vector is called the orthogonality constraint vector or abbreviated as the oc vector for the variable vector the matrix for the rayleigh quotient is an eigen balanced eb laplacian matrix a tighter lower bound of the constrained rayleigh quotient has many implications in mathematics as well as in cooperative control problems the main contributions in this paper are as follows first we provide the greatest lower bound or the infimum of the constrained rayleigh quotient with respect to a general oc vector and the eb laplacian matrix whose results are novel and better than the existing results then we interpret the physical meaning of our results in an insightful geometric form and characterize the properties of the results finally as an example to illustrate the merit of the tighter lower bound of the constrained rayleigh quotient we consider the scale of the agents driving by a fundamental consensus protocol and show that the convergence rate of the agents scale is characterized by the greatest lower bound of the constrained rayleigh quotient

a sharp lower bound for the least eigenvalue of the signless laplacian of a non bipartite graph we prove that the minimum value of the least eigenvalue of the signless laplacian of a connected non bipartite graph with a prescribed number of vertices is attained solely in the unicyclic graph obtained from a triangle by attaching a path at one of its endvertices

nonsmooth leader following formation control of nonidentical multi agent systems with directed communication topologies this paper presents two nonsmooth leader following formation protocols for nonidentical lipschitz nonlinear multi agent systems with directed communication network topologies one protocol is used to achieve finite time formation for first order systems and the other to achieve asymptotic formation for second order systems in these protocols the states of all the agents including the leader and the followers are available only locally within their neighborhoods some sufficient conditions for reaching formations are derived for nonidentical nonlinear systems satisfying locally lipschitz conditions to prove the stability a new nonsmooth lyapunov function is constructed with stability conditions derived under a nonsmooth analysis framework the proposed formation protocols are applied to multi spacecraft systems in deep space exploration with numerical simulations demonstrating the effectiveness of the theoretical results

consensus seeking in multiagent systems under dynamically changing interaction topologies this note considers the problem of information consensus among multiple agents in the presence of limited and unreliable information exchange with dynamically changing interaction topologies both discrete and continuous update schemes are proposed for information consensus this note shows that information consensus under dynamically changing interaction topologies can be achieved asymptotically if the union of the directed interaction graphs have a spanning tree frequently enough as the system evolves

distributed estimation and control of algebraic connectivity over random graphs in this paper we propose a distributed algorithm for the estimation and control of the connectivity of ad hoc networks in the presence of a random topology first given a generic random graph we introduce a novel stochastic power iteration method that allows each node to estimate and track the algebraic connectivity of the underlying expected graph using results from stochastic approximation theory we prove that the proposed method converges almost surely a s to the desired value of connectivity even in the presence of imperfect communication scenarios the estimation strategy is then used as a basic tool to adapt the power transmitted by each node of a wireless network in order to maximize the network connectivity in the presence of realistic medium access control mac protocols or simply to drive the connectivity toward a desired target value numerical results corroborate our theoretical findings thus illustrating the main features of the algorithm and its robustness to fluctuations of the network graph due to the presence of random link failures

the signless laplacian spectral radius of k connected irregular graphs abstract let g be a k connected irregular graph with n vertices m edges maximum degree and minimum degree in this paper we mainly show 2 q 1 g 2 n 2 m k 2 2 n 2 m n 2 2 k n 1 k 2 n k 2 where q 1 g is the signless laplacian spectral radius of g the inequality improves previous bounds of several authors in some cases it also implies a lower bound of 2 q 1 h for a proper subgraph h of a k connected regular graph another lower bound of 2 q 1 g for a connected graph g is also given

on the signless laplacian spectral radius of irregular graphs abstract let g be a connected irregular graph in this paper we give lower and upper bounds for the signless laplacian radius q 1 of g we show that q 1 4 e n 2 2 n and q 1 2 1 n d 1 4 hold where d is the diameter of g

a bounded distributed connectivity preserving aggregation strategy with collision avoidance property abstract this paper presents a bounded connectivity preserving control strategy for the aggregation of multi agent systems the problem is investigated for two cases of single integrator agents and unicycles under the proposed control strategy if two agents are in the connectivity range at some point in time they will stay connected thereafter the agents finally aggregate while avoiding collision in such a way that the average of the distances between every pair of neighboring agents is bounded by a pre specified positive real number which can be chosen arbitrarily small the results are developed based on some important characteristics of the positive limit set of the closed loop system under the proposed control strategy and a fundamental property of convex real functions the theoretical results are verified by simulation

static output feedback stabilization of interconnected systems abstract this paper is concerned with the celebrated static output feedback control problem sofp subject to linear constraints on control input k e g bounds on control magnitude zeros in some elements of k etc these constraints typically arise in the control of resource limited systems interconnected to each other where the local control for each system makes use of its own and neighboring systems outputs only and its magnitude is bounded this control problem can be approached by a spectral norm minimization technique q snm whose preliminary version was previously introduced with little mathematical justification but shown to be promising for the regular sofp without control constraints this paper mathematically justifies q snm by showing its explicit relationship with spectral radius and extends q snm to accommodate various linear control constraints this paper also discusses the practical application of q snm to vehicle formation control which demonstrates the merit of q snm

on two conjectures of randic index and the largest signless laplacian eigenvalue of graphs abstract the randic index r of a graph g is defined as the sum of d i d j 1 2 over all edges v i v j of g where d i denotes the degree of a vertex v i in g q 1 is the largest eigenvalue of the signless laplacian matrix q d a of g where d is the diagonal matrix with degrees of the vertices on the main diagonal and a is the adjacency matrix of g hansen and lucas 18 conjectured 1 q 1 r 3 2 n 2 and equality holds for g k n and 2 q 1 r 4 n 4 n 4 n 12 n n 1 n 13 with equality if and only if g k n for 4 n 12 and g s n for n 13 respectively in this paper we prove the conjecture 1 and obtain a result very close to the conjecture 2 moreover we give some results relating harmonic index and the largest eigenvalue of the adjacency matrix

on the singular second order elliptic equation abstract let m g be a smooth and compact riemannian manifold with dimensions n 3 this paper deals with the existence of nontrivial solutions for a nonlinear second order elliptic equation with a singular term and critical sobolev growth we provide a geometric application of this singular equation

which graphs are determined by their spectrum for almost all graphs the answer to the question in the title is still unknown here we survey the cases for which the answer is known not only the adjacency matrix but also other types of matrices such as the laplacian matrix are considered

bounds on the independence number and signless laplacian index of graphs abstract in this paper we give some bounds on the signless laplacian index of graphs in terms of independence number in addition these results disprove a conjecture in 3 involving the signless laplacian index and independence number of graphs

flocks and formations given a large number the flock of moving physical objects we investigate physically reasonable mechanisms of influencing their orbits in such a way that they move along a prescribed course and in a prescribed and fixed configuration or in formation each agent is programmed to see the position and velocity of a certain number of others this flow of information from one agent to another defines a fixed directed loopless graph in which the agents are represented by the vertices this graph is called the communication graph to be able to fly in formation an agent tries to match the mean position and velocity of his neighbors his direct antecedents on the communication graph to his own this operation defines a directed laplacian on the communication graph a linear feedback is used to ensure stability of the coherent flight patterns we analyze in detail how the connectedness of the communication graph affects the coherence of the stable flight patterns and give a characterization of these stable flight patterns we do the same if in addition the flight of the flock is guided by one or more leaders finally we use this theory to develop some applications examples of these are flight guided by external controls flocks of flocks and some results about flocks whose formation is always oriented along the line of flight such as geese

the second yamabe invariant let m g be a compact riemannian manifold of dimension n 3 we define the second yamabe invariant as the infimum of the second eigenvalue of the yamabe operator over the metrics conformal to g and of volume 1 we study when it is attained as an application we find nodal solutions of the yamabe equation

flocking with connectivity preservation of multiple double integrator systems subject to external disturbances by a distributed control law this paper studies the flocking with connectivity preservation problem of multiple double integrator systems subject to a class of external disturbances we solve the problem by a distributed dynamic position feedback control law under the standard assumptions our approach is a combination of the potential function technique and observer design our result also includes some existing results as special cases

time varying formation control for general linear multi agent systems with switching directed topologies time varying formation analysis and design problems for multi agent systems with general linear dynamics and switching directed interaction topologies are investigated different from the previous results the formation in this paper can be defined by specified piecewise continuously differentiable vectors and the switching topologies are directed firstly necessary and sufficient conditions for general linear multi agent systems with switching directed topologies to achieve time varying formations are proposed where a description of the feasible time varying formation set and approaches to expand the feasible formation set are given then an explicit expression of the time varying formation reference function is derived to describe the macroscopic movement of the whole formation an approach to assign the motion modes of the formation reference is provided moreover an algorithm consisting of four steps to design the formation protocol is presented in the case where the given time varying formation belongs to the feasible formation set it is proven that by designing the formation protocol using the proposed algorithm time varying formation can be achieved by multi agent systems with general linear dynamics and switching directed topologies if the dwell time is larger than a positive threshold finally numerical simulations are presented to demonstrate the effectiveness of the theoretical results

further results on the spectral radius of matrices and graphs abstract in this paper we obtain some results on the sharp upper bounds of the spectral radius of a nonnegative matrix then apply these results to signless laplacian matrices to obtain some known results we also apply these results to distance signless laplacian matrix of a graph and obtain some sharp bounds on the distance signless laplacian spectral radius