the spectrum and the signless laplacian spectrum of coronae abstract let g 1 g 2 be two simple connected graphs denote the corona and the edge corona of g 1 g 2 by g 1 g 2 and g 1 g 2 respectively in this paper we first introduce a new invariant the m coronal of a graph matrix m where the matrix m is associated with a graph in a prescribed way then it is used to compute the signless laplacian spectrum of g 1 g 2 and g 1 g 2 in terms of the signless laplacian spectrum of g 1 and g 2 in addition the spectrum of g 1 g 2 are also given in terms of the spectrum of g 1 and g 2 finally as an application of these results we construct many pairs of nonisomorphic signless laplacian cospectral graphs

energy of line graphs the energy of a graph is equal to the sum of the absolute values of its eigenvalues the energy of a matrix is equal to the sum of its singular values we establish relations between the energy of the line graph of a graph g and the energies associated with the laplacian and signless laplacian matrices of g

maximizing the least signless laplacian eigenvalue of unicyclic graphs abstract in this paper a graph with the maximum least signless laplacian eigenvalue among all connected unicyclic graphs with fixed order is determined

trees with minimal laplacian coefficients let g be a simple undirected graph with the characteristic polynomial of its laplacian matrix l g p g m email protected k 0 n 1 kc email protected n k it is well known that for trees the laplacian coefficient c n 2 is equal to the wiener index of g while c n 3 is equal to the modified hyper wiener index of the graph in this paper we characterize n vertex trees with given matching number m which simultaneously minimize all laplacian coefficients the extremal tree a n m is a spur obtained from the star graph s n m 1 with n m 1 vertices by attaching a pendant edge to each of certain m 1 non central vertices of s n m 1 in particular a n m minimizes the wiener index the modified hyper wiener index and the recently introduced incidence energy of trees defined as ie g email protected k 0 email protected k where m k are the eigenvalues of signless laplacian matrix q g d g a g we introduced a general r transformation which decreases all laplacian coefficients simultaneously in conclusion we illustrate on examples of wiener index and incidence energy that the opposite problem of simultaneously maximizing all laplacian coefficients has no solution

mean and variance of implicitly defined biased estimators such as penalized maximum likelihood applications to tomography many estimators in signal processing problems are defined implicitly as the maximum of some objective function examples of implicitly defined estimators include maximum likelihood penalized likelihood maximum a posteriori and nonlinear least squares estimation for such estimators exact analytical expressions for the mean and variance are usually unavailable therefore investigators usually resort to numerical simulations to examine the properties of the mean and variance of such estimators this paper describes approximate expressions for the mean and variance of implicitly defined estimators of unconstrained continuous parameters we derive the approximations using the implicit function theorem the taylor expansion and the chain rule the expressions are defined solely in terms of the partial derivatives of whatever objective function one uses for estimation as illustrations we demonstrate that the approximations work well in two tomographic imaging applications with poisson statistics we also describe a plug in approximation that provides a remarkably accurate estimate of variability even from a single noisy poisson sinogram measurement the approximations should be useful in a wide range of estimation problems

l infty exact observability of the heat equationwith scanning pointwise sensor the problem of exact observability of the heat equation in an arbitrary space dimension with scanning pointwise sensor is considered in the case when the space for outputs is l infty varepsilon theta varepsilon 0 a new method for the construction of observation curves for sensors that are able to ensure l infty varepsilon theta exact observability at final time is given based on the maximum principle for the heat equation an application of the method to the observability problem with discrete time scanning observations and related approximate controllability results are also discussed

the smallest eigenvalue of the signless laplacian recently the signless laplacian matrix of graphs has been intensively investigated while there are many results about the largest eigenvalue of the signless laplacian the properties of its smallest eigenvalue are less well studied the present paper surveys the known results and presents some new ones about the smallest eigenvalue of the signless laplacian

on the nullity of the line graph of unicyclic graph with depth one abstract a connected graph with a unique cycle is called a unicyclic graph a unicyclic graph with depth one may be thought of as being obtained from a cycle by appending n i pendent edges on each vertex v i in the cycle c t for some integer t 3 denoted by c n 1 n 2 n t in this paper we give a complete characterization on the nullity of the line graph g l c n 1 n 2 n t as follows let m i n i 0 then i g 2 if and only if m 0 and t 0 mod 4 ii g 1 if and only if m 1 and either 1 n i 0 1 for i 1 t the length of any zero chain of n 1 n 2 n t is even and t m 0 mod 4 or 2 t 0 mod 4 and one of n 1 n 3 n t 1 0 and n 2 n 4 n t 0 must hold iii g 0 otherwise

on the reduced signless laplacian spectrum of a degree maximal graph abstract for a simple graph g the signless laplacian of g is the matrix a g d g where a g is the adjacency matrix and d g is the diagonal matrix of vertex degrees of g the reduced signless laplacian of g is the matrix g b g where b g is the reduced adjacency matrix of g and g is the diagonal matrix whose diagonal entries are the common degrees for vertices belonging to the same neighborhood equivalence class of g a graph is said to be degree maximal if it is connected and its degree sequence is not majorized by the degree sequence of any other connected graph for a maximal graph we obtain a formula for the characteristic polynomial of its reduced signless laplacian and use the formula to derive a localization result for its reduced signless laplacian eigenvalues and to compare the signless laplacian spectral radii of two well known maximal graphs we also obtain a necessary condition for a maximal graph to have maximal signless laplacian spectral radius among all connected graphs with given numbers of vertices and edges

on the laplacian and signless laplacian spectrum of a graph with k pairwise co neighbor vertices abstract consider the laplacian and signless laplacian spectrum of a graph g of order n with k pairwise co neighbor vertices we prove that the number of shared neighbors is a laplacian and a signless laplacian eigenvalue of g with multiplicity at least k 1 additionally considering a connected graph g k with a vertex set defined by the k pairwise co neighbor vertices of g the laplacian spectrum of g k obtained from g adding the edges of g k includes l for each nonzero laplacian eigenvalue of g k the laplacian spectrum of g overlaps the laplacian spectrum of g k in at least n k 1 places

sharp bounds for the largest eigenvalue of the signless laplacian of a graph let g be a simple connected graph with n vertices and m edges denote the degree of vertex vi by d vi the matrix q g d g a g is called the signless laplacian of g where d g diag d v1 d v2 d vn and a g denote the diagonal matrix of vertex degrees and the adjacency matrix of g respectively let q1 g be the largest eigenvalue of q g in this paper we first present two sharp upper bounds for q1 g involving the maximum degree and the minimum degree of the vertices of g and give a new proving method on another sharp upper bound for q1 g then we present three sharp lower bounds for q1 g involving the maximum degree and the minimum degree of the vertices of g moreover we determine all extremal graphs which attain these sharp bounds

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

bounds for different spreads of line and total graphs abstract in this paper we explore some results concerning the spread of the line and the total graph of a given graph a sufficient condition for the spread of a unicyclic graph with an odd girth to be at most the spread of its line graph is presented additionally we derive an upper bound for the spread of the line graph of graphs on n vertices having a vertex edge connectivity at most a positive integer k combining techniques of interlacing of eigenvalues we derive lower bounds for the laplacian and signless laplacian spread of the total graph of a connected graph moreover for a regular graph an upper and lower bound for the spread of its total graph is given

penalized likelihood reconstruction with high fidelity measurement models for high resolution cone beam imaging we present a novel reconstruction algorithm based on a general cone beam ct forward model which is capable of incorporating the blur and noise correlations that are exhibited in flat panel cbct measurement data specifically the proposed model may include scintillator blur focal spot blur and noise correlations due to light spread in the scintillator the proposed algorithm gpl bc uses a gaussian penalized likelihood objective function which incorporates models of blur and correlated noise in a simulation study gpl bc was able to achieve lower bias as compared with deblurring followed by fdk as well as a model based reconstruction method without integration of measurement blur in the same study gpl bc was able to achieve better line pair reconstructions in terms of segmented image accuracy as compared with deblurring followed by fdk a model based method without blur and a model based method with blur but not noise correlations a prototype extremities quantitative cone beam ct test bench was used to image a physical sample of human trabecular bone these data were used to compare reconstructions using the proposed method and model based methods without blur and or correlation to a registered mu ct image of the same bone sample the gpl bc reconstructions resulted in more accurate trabecular bone segmentation multiple trabecular bone metrics including trabecular thickness tb th were computed for each reconstruction approach as well as the mu ct volume the gpl bc reconstruction provided the most accurate tb th measurement 0 255 mm as compared with the mu ct derived value of 0 193 mm followed by the gpl b reconstruction the gpl i reconstruction and then the fdk reconstruction 0 271 mm 0 309 mm and 0 335 mm respectively

on eigenspaces of some compound signed graphs abstract in the theory of simple graphs the concepts of the line and subdivision graph as compound graphs are well known it is possible to consider them also in the context of edge signed graphs some relations between the laplacian spectrum of signed graphs and adjacency spectra of their associated compound signed graphs have been recently established in the literature in this paper we study the relations between the corresponding eigenspaces

starlike trees whose maximum degree exceed 4 are determined by their q spectra abstract let a g and d g be the adjacency matrix and the degree matrix of a graph g respectively the matrix d g a g is called the signless laplacian matrix of g the spectrum of the matrix d g a g is called the q spectrum of g a graph is said to be determined by its q spectrum if there is no other non isomorphic graph with the same q spectrum in this paper we prove that all starlike trees whose maximum degree exceed 4 are determined by their q spectra

graphs with least eigenvalue 2 ten years on abstract the authors monograph spectral generalizations of line graphs was published in 2004 following the successful use of star complements to complete the classification of graphs with least eigenvalue 2 guided by citations of the book we survey progress in this area over the past decade some new observations are included

on some forests determined by their laplacian or signless laplacian spectrum we consider the class of graphs whose each component is either a proper subgraph of some smith graphs or belongs to a precized subset of smith graphs we classify the graphs from the considered class into those which are determined or not determined by laplacian or signless laplacian spectrum

signless laplacian spectral radii of graphs with given chromatic number abstract let g be a simple graph with vertices v 1 v 2 v n of degrees d 1 d 2 d n respectively let a be the 0 1 adjacency matrix of g and d be the diagonal matrix diag d 1 d 2 d n q g d a is called the signless laplacian of g the largest eigenvalue of q g is called the signless laplacian spectral radius or q spectral radius of g denote by g the chromatic number for a graph g in this paper for graphs with order n the extremal graphs with both the given chromatic number and the maximal q spectral radius are characterized the extremal graphs with both the given chromatic number 4 5 6 7 and the minimal q spectral radius are characterized as well

maximum likelihood transmission image reconstruction for overlapping transmission beams in many transmission imaging geometries the transmitted beams of photons overlap on the detector such that a detector element may record photons that originated in different sources or source locations and thus traversed different paths through the object examples include systems based on scanning line sources or on multiple parallel rod sources the overlap of these beams has been disregarded by both conventional analytical reconstruction methods as well as by previous statistical reconstruction methods the authors propose a new algorithm for statistical image reconstruction of attenuation maps that explicitly accounts for overlapping beams in transmission scans the algorithm is guaranteed to monotonically increase the objective function at each iteration the availability of this algorithm enables the possibility of deliberately increasing the beam overlap so as to increase count rates simulated single photon emission tomography transmission scans based on a multiple line source array demonstrate that the proposed method yields improved resolution noise tradeoffs relative to conventional reconstruction algorithms both statistical and nonstatistical

on products and line graphs of signed graphs their eigenvalues and energy abstract in this article we examine the adjacency and laplacian matrices and their eigenvalues and energies of the general product non complete extended p sum or neps of signed graphs we express the adjacency matrix of the product in terms of the kronecker matrix product and the eigenvalues and energy of the product in terms of those of the factor graphs for the cartesian product we characterize balance and compute expressions for the laplacian eigenvalues and laplacian energy we give exact results for those signed planar cylindrical and toroidal grids which are cartesian products of signed paths and cycles we also treat the eigenvalues and energy of the line graphs of signed graphs and the laplacian eigenvalues and laplacian energy in the regular case with application to the line graphs of signed grids that are cartesian products and to the line graphs of all positive and all negative complete graphs

the least eigenvalue of signless laplacian of graphs under perturbation abstract we investigate how the least eigenvalue of the signless laplacian of a graph changes by relocating a bipartite branch from one vertex to another vertex and minimize the least eigenvalue of the signless laplacian among the class of connected graphs with fixed order which contains a given non bipartite graph as an induced subgraph

optimal measurement trajectories for distributed parameter systems abstract the paper deals with the optimal sensor location problem associated with minimax filtering for linear distributed parameter systems under the moving sensors the problem of optimal choice of a measurement trajectory is treated here as an h optimal control one under phase constraints with the objective to minimize a given weak performance index under worst possible disturbances existence of a solution to such a problem is established for the case of quadratic constraints on the disturbances and necessary conditions for optimally are derived on the basis of constructing a sequence of suboptimal solutions associated in turn with a sequence of finite dimensional maximum principles

sensors and controllers location in distributed systems a survey a survey of the field of optimal sensors and or controllers location for dynamical distributed parameter systems modelled by partial differential equations is presented the recent contributions in this field are grouped according to the main goal for which the location problem is developed namely system identification state estimation and optimal control in order to pose the sensors and controllers location problem the semigroup approach for modelling distributed linear systems is briefly reviewed together with its equivalent infinite dimensional and approximate finite dimensional fourier expansion representations after presenting a concise general review of the several methods considered in the current literature a classification of methods is also proposed the main classifying factor concerns the use of n modal approximation schemes and the different stages of the optimization procedure in which they are required

on the laplacian coefficients of signed graphs abstract let g be a signed graph where g is its underlying graph and its sign function defined on edges of g a signed graph the subgraph of is its signed tu subgraph if the signed graph induced by the vertices of consists of trees and or unbalanced unicyclic signed graphs let l d g a be the laplacian of in this paper we express the coefficient of the laplacian characteristic polynomial of based on the signed tu subgraphs of and establish the relation between the laplacian characteristic polynomial of a signed graph with adjacency characteristic polynomials of its signed line graph and signed subdivision graph as an application we identify the signed unicyclic graphs having extremal coefficients of the laplacian characteristic polynomial

on the z eigenvalues of the adjacency tensors for uniform hypergraphs abstract the adjacency matrices for graphs are generalized to the adjacency tensors for uniform hypergraphs and some fundamental properties for the adjacency tensor and its z eigenvalues of a uniform hypergraph are obtained in particular some bounds on the smallest and the largest z eigenvalues of the adjacency tensors for uniform hypergraphs are presented

the deformed consensus protocol this paper studies a generalization of the standard continuous time consensus protocol obtained by replacing the laplacian matrix of the communication graph with the so called deformed laplacian the deformed laplacian is a second degree matrix polynomial in the real variable s which reduces to the standard laplacian for s equal to unity the stability properties of the ensuing deformed consensus protocol are studied in terms of parameter s for some special families of undirected and directed graphs and for arbitrary graph topologies by leveraging the spectral theory of quadratic eigenvalue problems examples and simulation results are provided to illustrate our theoretical findings

the least signless laplacian eigenvalue of non bipartite graphs with given stability number abstract in this paper we investigate the least eigenvalue of signless laplacian matrix of a connected graph and determine the graphs which have the minimum least signless laplacian eigenvalue among all nonbipartite graphs with given stability number n 1 2 or covering number 2 n 1 2 respectively

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

first eigenvalue of nonsingular mixed graphs with given number of pendant vertices abstract let g be a mixed graph and l g be the laplacian matrix of the mixed graph g the first eigenvalue of g is referred to the least nonzero eigenvalue of l g let mg n k be the set of nonsingular mixed graphs with n vertices and k pendant vertices where n 4 in this paper up to a signature matrix we determine the unique graph with the minimal first eigenvalue among all graphs in m g n k thus we obtain a lower bound for the first eigenvalue of a mixed graph in terms of the number of pendant vertices

an expanded theoretical treatment of iteration dependent majorize minimize algorithms the majorize minimize mm optimization technique has received considerable attention in signal and image processing applications as well as in statistics literature at each iteration of an mm algorithm one constructs a tangent majorant function that majorizes the given cost function and is equal to it at the current iterate the next iterate is obtained by minimizing this tangent majorant function resulting in a sequence of iterates that reduces the cost function monotonically a well known special case of mm methods are expectation maximization algorithms in this paper we expand on previous analyses of mm due to fessler and hero that allowed the tangent majorants to be constructed in iteration dependent ways also this paper overcomes an error in one of those earlier analyses there are three main aspects in which our analysis builds upon previous work first our treatment relaxes many assumptions related to the structure of the cost function feasible set and tangent majorants for example the cost function can be nonconvex and the feasible set for the problem can be any convex set second we propose convergence conditions based on upper curvature bounds that can be easier to verify than more standard continuity conditions furthermore these conditions allow for considerable design freedom in the iteration dependent behavior of the algorithm finally we give an original characterization of the local region of convergence of mm algorithms based on connected e g convex tangent majorants for such algorithms cost function minimizers will locally attract the iterates over larger neighborhoods than typically is guaranteed with other methods this expanded treatment widens the scope of the mm algorithm designs that can be considered for signal and image processing applications allows us to verify the convergent behavior of previously published algorithms and gives a fuller understanding overall of how these algorithms behave

the domination number and the least q eigenvalue abstract in this paper for a nonbipartite graph with both order n and domination number we show that it contains a nonbipartite unicyclic spanning subgraph with the same domination number we also present some results about the changing of the domination number when the structure of a graph is changed by investigating the relation between the domination number and the least q eigenvalue of a graph we minimize the least q eigenvalue among all the nonbipartite graphs with both order n 4 and given domination number n 1 3

asymptotic normality of laplacian coefficients of graphs abstract let g be a simple graph with n vertices and let c g x k 0 n 1 n k c g k x k denote the laplacian characteristic polynomial of g then if the size e g is large compared to the maximum degree g laplacian coefficients c g k are approximately normally distributed by central and local limit theorems we show that laplacian coefficients of the paths the cycles the stars the wheels and regular graphs of degree d are approximately normally distributed respectively we also point out that laplacian coefficients of the complete graphs and the complete bipartite graphs are approximately poisson distributed respectively

spectral properties of complex unit gain graphs abstract a complex unit gain graph is a graph where each orientation of an edge is given a complex unit which is the inverse of the complex unit assigned to the opposite orientation we extend some fundamental concepts from spectral graph theory to complex unit gain graphs we define the adjacency incidence and laplacian matrices and study each of them the main results of the paper are eigenvalue bounds for the adjacency and laplacian matrices

monotonic algorithms for transmission tomography we present a framework for designing fast and monotonic algorithms for transmission tomography penalized likelihood image reconstruction the new algorithms are based on paraboloidal surrogate functions for the log likelihood due to the form of the log likelihood function it is possible to find low curvature surrogate functions that guarantee monotonicity unlike previous methods the proposed surrogate functions lead to monotonic algorithms even for the nonconvex log likelihood that arises due to background events such as scatter and random coincidences the gradient and the curvature of the likelihood terms are evaluated only once per iteration since the problem is simplified at each iteration the cpu time is less than that of current algorithms which directly minimize the objective yet the convergence rate is comparable the simplicity monotonicity and speed of the new algorithms are quite attractive the convergence rates of the algorithms are demonstrated using real and simulated pet transmission scans

controllability of undirected graphs abstract in control theory networked dynamical systems have a wide range of engineering applications in a relational graph among followers f and leaders r new necessary and sufficient conditions for a pair f r to be controllable are presented the choice of leader vertices for controllability is shown to be facilitated by identifying the core vertices associated with the eigenvectors of a matrix s related to a graph we present new necessary and sufficient conditions for a graph to be controllable relative to its adjacency matrix or to its signless laplacian without having to evaluate any eigenspaces which is the criterion usually employed the symmetries of the system graph represented by s are also shown to aid in the choice of a potential leader vertex that is able to control the follower subgraph on its own moreover we define k omnicontrollable graphs for controllability by any k leaders and show that simple 1 omnicontrollable graphs have only two possible automorphism groups

signed graphs with extremal least laplacian eigenvalue abstract a signed graph is a pair g where g v g e g is a graph and e g is the corresponding sign function for a signed graph we consider the laplacian matrix defined as l d g a where d g is the matrix of vertex degrees of g and a is the signed adjacency matrix it is well known that is balanced that is each cycle contains an even number of negative edges if and only if the least laplacian eigenvalue n 0 therefore if is not balanced then n 0 we show here that among unbalanced connected signed graphs of given order the least eigenvalue is minimal for an unbalanced triangle with a hanging path while the least eigenvalue is maximal for the complete graph with the all negative sign function

minimizing the least eigenvalues of unicyclic graphs with application to spectral spread the spread of a graph is defined to be the difference between the largest eigenvalue and the least eigenvalue of the adjacency matrix of the graph let u k denote the set of connected unicyclic graphs of order n and girth k and let un denote the set of connected unicyclic graphs of order n in this paper we determine the unique graph with minimum least eigenvalue respectively the unique graph with maximum spread among all graphs in u k we finally characterize the unique graph with minimum least eigenvalue respectively the unique graph with maximum spread among all graphs in un

globally convergent algorithms for maximum a posteriori transmission tomography this paper reviews and compares three maximum likelihood algorithms for transmission tomography one of these algorithms is the em algorithm one is based on a convexity argument devised by de pierro see ieee trans med imaging vol 12 p 328 333 1993 in the context of emission tomography and one is an ad hoc gradient algorithm the algorithms enjoy desirable local and global convergence properties and combine gracefully with bayesian smoothing priors preliminary numerical testing of the algorithms on simulated data suggest that the convex algorithm and the ad hoc gradient algorithm are computationally superior to the em algorithm this superiority stems from the larger number of exponentiations required by the em algorithm the convex and gradient algorithms are well adapted to parallel computing

the minimum signless laplacian spectral radius of graphs with given independence number abstract the independence number g of g is defined as the maximum cardinality of a set of pairwise non adjacent vertices which is called an independent set in this paper we characterize the graphs which have the minimum spectral radius among all the connected graphs of order n with independence number g 1 2 n 2 n 2 1 n 2 n 3

perturbations of discrete elliptic operators abstract given v a finite set a self adjoint operator k on c v is called elliptic if it is positive semi definite and its lowest eigenvalue is simple therefore there exists a unique up to sign unitary function c v satisfying k and then k is named elliptic clearly a elliptic operator is singular iff 0 examples of elliptic operators are the so called schrodinger operators on finite connected networks as well as the signless laplacian of connected bipartite networks a elliptic operator k defines an automorphism on whose inverse is called orthogonal green operator of k we aim here at studying the effect of a perturbation of k on its orthogonal green operator the perturbation here considered is performed by adding a self adjoint and positive semi definite operator to k as particular cases we consider the effect of changing the conductances on semi definite schodinger operators on finite connected networks and on the signless laplacian of connected bipartite networks the expression obtained for the perturbed network is explicitly given in terms of the orthogonal green function of the original network

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

maximizing signless laplacian or adjacency spectral radius of graphs subject to fixed connectivity in this paper we characterize the graphs with maximum signless laplacian or adjacency spectral radius among all graphs with fixed order and given vertex or edge connectivity we also discuss the minimum signless laplacian or adjacency spectral radius of graphs subject to fixed connectivity consequently we give an upper bound of signless laplacian or adjacency spectral radius of graphs in terms of connectivity in addition we confirm a conjecture of aouchiche and hansen involving adjacency spectral radius and connectivity

on graphs with exactly three q eigenvalues at least two abstract for a graph g we here investigate its signless laplacian matrix q g and the corresponding q eigenvalues by considering the relation between the q spectrum and the circumference of g we characterize all connected graphs with exactly three q eigenvalues at least two

on the extremal values of the second largest q eigenvalue we study extremal graphs for the extremal values of the second largest q eigenvalue of a connected graph we first characterize all simple connected graphs with second largest signless laplacian eigenvalue at most 3 the second part of the present paper is devoted to the study of the graphs that maximize the second largest q eigenvalue we construct families of such graphs and prove that some of theses families are minimal for the fact that they maximize the second largest signless laplacian eigenvalue

globally convergent edge preserving regularized reconstruction an application to limited angle tomography we introduce a generalization of a deterministic relaxation algorithm for edge preserving regularization in linear inverse problems this algorithm transforms the original possibly nonconvex optimization problem into a sequence of quadratic optimization problems and has been shown to converge under certain conditions when the original cost functional being minimized is strictly convex we prove that our more general algorithm is globally convergent i e converges to a local minimum from any initialization under less restrictive conditions even when the original cost functional is nonconvex we apply this algorithm to tomographic reconstruction from limited angle data by formulating the problem as one of regularized least squares optimization the results demonstrate that the constraint of piecewise smoothness applied through the use of edge preserving regularization can provide excellent limited angle tomographic reconstructions two edge preserving regularizers one convex the other nonconvex are used in numerous simulations to demonstrate the effectiveness of the algorithm under various limited angle scenarios and to explore how factors such as the choice of error norm angular sampling rate and amount of noise affect the reconstruction quality and algorithm performance these simulation results show that for this application the nonconvex regularizer produces consistently superior results

theorems on partitioned matrices revisited and their applications to graph spectra some old results about spectra of partitioned matrices due to goddard and schneider or haynsworth are re proved a new result is given for the spectrum of a block stochastic matrix with the property that each off diagonal block has equal entries and each diagonal block has equal diagonal entries and equal off diagonal entries the result is applied to the study of the spectra of the usual graph matrices by partitioning the vertex set of the graph according to the neighborhood equivalence relation the concept of a reduced graph matrix is introduced the question of when n 2 is the second largest signless laplacian eigenvalue of a connected graph of order n is treated a recent conjecture posed by tam fan and zhou on graphs that maximize the signless laplacian spectral radius over all not necessarily connected graphs with given numbers of vertices and edges is refuted the laplacian spectrum of a degree maximal graph is reconsidered

a local update strategy for iterative reconstruction from projections a method for bayesian reconstruction which relies on updates of single pixel values rather than the entire image at each iteration is presented the technique is similar to gauss seidel gs iteration for the solution of differential equations on finite grids the computational cost per iteration of the gs approach is found to be approximately equal to that of gradient methods for continuously valued images gs is found to have significantly better convergence at modes representing high spatial frequencies in addition gs is well suited to segmentation when the image is constrained to be discretely valued it is shown that bayesian segmentation using gs iteration produces useful estimates at much lower signal to noise ratios than required for continuously valued reconstruction the convergence properties of gradient ascent and gs for reconstruction from integral projections are analyzed and simulations of both maximum likelihood and maximum a posteriori cases are included

on the least q eigenvalue of a non bipartite hamiltonian graph abstract in this paper we study the least q eigenvalue of a non bipartite hamiltonian graph and determine all non bipartite hamiltonian graphs whose least q eigenvalue attains the minimum among all non bipartite hamiltonian graphs on n vertices

the signless laplacian coefficients and incidence energy of bicyclic graphs abstract let q g x det x i q g i 1 n 1 i i x n i be the characteristic polynomial of the signless laplacian matrix of a graph g of order n this paper investigates how the signless laplacian coefficients i e coefficients of q g x change after some graph transformations these results are used to prove that the set b n of all bicyclic graphs of order n has exactly two minimal elements with respect to the partial ordering of their coefficients furthermore we present a sharp lower bound for the incidence energy of bicyclic graphs of order n and characterize all extremal graphs

characterizing threshold graphs with k main signless laplacian eigenvalues abstract we show that the number k of main signless laplacian eigenvalues of a threshold graph on n 3 vertices is bounded by n 2 and we characterize for any integer 1 k n 2 all such graphs with exactly k main signless laplacian eigenvalues moreover we also determine the number of threshold graphs on n vertices with k main signless laplacian eigenvalues

connected graphs with maximal q index the one dominating vertex case abstract by the signless laplacian of a simple graph g we mean the matrix q g d g a g where a g d g denote respectively the adjacency matrix and the diagonal matrix of vertex degrees of g for every pair of positive integers n k it is proved that if 3 k n 3 then h n k the graph obtained from the star k 1 n 1 by joining a vertex of degree 1 to k 1 other vertices of degree 1 is the unique connected graph that maximizes the largest signless laplacian eigenvalue over all connected graphs with n vertices and n k edges

unoriented laplacian maximizing graphs are degree maximal abstract a connected graph is said to be unoriented laplacian maximizing if the spectral radius of its unoriented laplacian matrix attains the maximum among all connected graphs with the same number of vertices and the same number of edges a graph is said to be threshold maximal if its degree sequence is not majorized by the degree sequence of any other graph and in addition the graph is connected it is proved that an unoriented laplacian maximizing graph is maximal and also that there are precisely two unoriented laplacian maximizing graphs of a given order and with nullity 3 our treatment depends on the following known characterization a graph g is threshold maximal if and only if for every pair of vertices u v of g the sets n u v n v u where n u denotes the neighbor set of u in g are comparable with respect to the inclusion relation and in addition the graph is connected a conjecture about graphs that maximize the unoriented laplacian matrix among all graphs with the same number of vertices and the same number of edges is also posed

optimum choice of moving sensor trajectories for distributed parameter system identification the problem is considered of the optimal choice of moving sensor trajectories from the viewpoint of distributed parameter system identification accuracy the determinant of the information matrix of parameter estimates is taken as a measure of identification accuracy necessary and sufficient conditions for the trajectories to be optimal are derived it is shown that the optimum trajectories can be found by solving a sequence of optimal sensor allocation problems a number of examples illustrates the proposed approach

proof of conjecture involving the second largest signless laplacian eigenvalue and the index of graphs abstract let g v e be a simple graph denote by d g the diagonal matrix of its vertex degrees and by a g its adjacency matrix then the signless laplacian matrix of g is q g d g a g in 5 cvetkovic et al have given the following conjecture involving the second largest signless laplacian eigenvalue q 2 and the index 1 of graph g see also aouchiche and hansen 1 1 n 1 q 2 1 n 2 2 n 4 with equality if and only if g is the star k 1 n 1 for the lower bound and if and only if g is the complete bipartite graph k n 2 2 for the upper bound in this paper we prove the lower bound and characterize the extremal graphs

spectral characterizations of signed lollipop graphs abstract let g be a signed graph where g is the underlying simple graph and e g is the sign function on the edges of g in this paper we consider the spectral characterization problem extended to the adjacency matrix and laplacian matrix of signed graphs after giving some basic results we study the spectral determination of signed lollipop graphs and we show that any signed lollipop graph is determined by the spectrum of its laplacian matrix

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

laplacian coefficients of trees with given number of leaves or vertices of degree two abstract let g be a simple undirected n vertex graph with the characteristic polynomial of its laplacian matrix l g det i l g k 0 n 1 k c k n k it is well known that for trees the laplacian coefficient c n 2 is equal to the wiener index of g while c n 3 is equal to the modified hyper wiener index of graph using a result of zhou and gutman on the relation between the laplacian coefficients and the matching numbers in subdivided bipartite graphs we characterize the trees with k leaves pendent vertices which simultaneously minimize all laplacian coefficients in particular this extremal balanced starlike tree s n k minimizes the wiener index the modified hyper wiener index and recently introduced laplacian like energy we prove that graph s n n 1 p has minimal laplacian coefficients among n vertex trees with p vertices of degree two in conclusion we illustrate on examples of these spectrum based invariants that the opposite problem of simultaneously maximizing all laplacian coefficients has no solution and pose a conjecture on extremal unicyclic graphs with k leaves

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

fast gradient based methods for bayesian reconstruction of transmission and emission pet images the authors describe conjugate gradient algorithms for reconstruction of transmission and emission pet images the reconstructions are based on a bayesian formulation where the data are modeled as a collection of independent poisson random variables and the image is modeled using a markov random field a conjugate gradient algorithm is used to compute a maximum a posteriori map estimate of the image by maximizing over the posterior density to ensure nonnegativity of the solution a penalty function is used to convert the problem to one of unconstrained optimization preconditioners are used to enhance convergence rates these methods generally achieve effective convergence in 15 25 iterations reconstructions are presented of an sup 18 fdg whole body scan from data collected using a siemens cti ecat931 whole body system these results indicate significant improvements in emission image quality using the bayesian approach in comparison to filtered backprojection particularly when reprojections of the map transmission image are used in place of the standard attenuation correction factors

the least q eigenvalue with fixed domination number abstract denote by lg l the lollipop graph obtained by attaching a pendant path p v g v g 1 v g l l 1 to a cycle c v 1 v 2 v g v 1 g 3 a f g l g r a p h of order n g 1 is defined to be the graph obtained by attaching n g l pendent vertices to some of the nonpendant vertices of lg l in which each vertex other than v g l 1 is attached at most one pendant vertex a f g l graph is a f g l g r a p h in which vg is attached with pendant vertex denote by qmin the least q e i g e n v a l u e of a graph in this paper we proceed on considering the domination number the least q eigenvalue of a graph as well as their relation further results obtained are as follows i some results about the changing of the domination number under the structural perturbation of a graph are represented ii among all nonbipartite unicyclic graphs of order n with both domination number and girth g g n 1 the minimum qmin attains at a f g l graph for some l iii among the nonbipartite graphs of order n and with given domination number which contain a f g l graph as a subgraph some lower bounds for qmin are represented iv among the nonbipartite graphs of order n and with given domination number n 2

permanental roots and the star degree of a graph abstract for undirected graphs without loops or multiple edges we define the star degree of a graph and prove that it is equal to the multiplicity of the root 1 of per xi b where b d a considering bipartite graphs we prove that per xi b per xi l where l d a and consequently that the star degree of a bipartite graph can also be characterized by the multiplicity of the root 1 of per xi l

signless laplacian spectral radius and hamiltonicity we give tight conditions on the signless laplacian spectral radius of a graph for the existence of hamiltonian paths and cycles

a modified expectation maximization algorithm for penalized likelihood estimation in emission tomography the maximum likelihood ml expectation maximization em approach in emission tomography has been very popular in medical imaging for several years in spite of this no satisfactory convergent modifications have been proposed for the regularized approach here a modification of the em algorithm is presented the new method is a natural extension of the em for maximizing likelihood with concave priors convergence proofs are given

generalized em type reconstruction algorithms for emission tomography we provide a general form for many reconstruction estimators of emission tomography these estimators include shepp and vardi s maximum likelihood ml estimator the quadratic weighted least squares wls estimator anderson s wls estimator and liu and wang s multi objective estimator and others we derive a generic update rule by constructing a surrogate function this work is inspired by the ml em em expectation maximization where the latter naturally arises as a special case a regularization with a specific form can also be incorporated by de pierro s trick we provide a general and quite different convergence proof compared with the proofs of the ml em and de pierro theoretical analysis shows that the proposed algorithm monotonically decreases the cost function and automatically meets nonnegativity constraints we have introduced a mechanism to provide monotonic self constraining and convergent algorithms from which some interesting existing and new algorithms can be derived simulation results illustrate the behavior of these algorithms in term of image quality and resolution noise tradeoff

extremizing first eigenvalue of 3 colored digraphs made with given blocks abstract this article describes the structure of the non singular connected 3 colored digraphs extremizing the first eigenvalue among all such graphs made with some given blocks as an application we determine the non singular unicyclic 3 colored digraph which minimizes the first eigenvalue over all such graphs with fixed order and girth

spectra of copies of a generalized bethe tree attached to any graph abstract a generalized bethe tree is a rooted unweighted tree in which vertices at the same level have the same degree let g be any connected graph let g b be the graph obtained from g by attaching a generalized bethe tree b by its root to each vertex of g we characterize completely the eigenvalues of the signless laplacian laplacian and adjacency matrices of the graph g b including results on the eigenvalue multiplicities finally for the laplacian and signless laplacian matrices we recall a procedure to compute a tight upper bound on the algebraic connectivity of g b as well as on the smallest eigenvalue of the signless laplacian matrix of g b whenever g is a non bipartite graph

on the signless laplacian spectral radius of graphs with cut vertices in this paper we show that among all the connected graphs with n vertices and k cut vertices the maximal signless laplacian spectral radius is attained uniquely at the graph gn k where gn k is obtained from the complete graph kn k by attaching paths of almost equal lengths to all vertices of kn k we also give a new proof of the analogous result for the spectral radius of the connected graphs with n vertices and k cut vertices see a berman x d zhang on the spectral radius of graphs with cut vertices j combin theory ser b 83 2001 233 240 finally we discuss the limit point of the maximal signless laplacian spectral radius