two laplacians for the distance matrix of a graph abstract we introduce a laplacian and a signless laplacian for the distance matrix of a connected graph called the distance laplacian and distance signless laplacian respectively we show the equivalence between the distance signless laplacian distance laplacian and the distance spectra for the class of transmission regular graphs there is also an equivalence between the laplacian spectrum and the distance laplacian spectrum of any connected graph of diameter 2 similarities between n as a distance laplacian eigenvalue and the algebraic connectivity are established

diffusions for global optimizations we seek a global minimum of u 0 1 n to r the solution to d dt x t nabla u x t will find local minima the solution to dx t nabla u x t dt sqrt 2t dw t where w is standard n dimensional brownian motion and the boundaries are reflecting will concentrate near the global minima of u at least when temperature t is small the equilibrium distribution for x t is gibbs with density pi t x alpha exp u x t this suggests setting t t t downarrow 0 to find the global minima of u we give conditions on u x and t t such that the solution to dx t nabla u x t dt sqrt 2t dw t converges weakly to a distribution concentrated on the global minima of u

distance spectra of graphs a survey abstract in 1971 graham and pollack established a relationship between the number of negative eigenvalues of the distance matrix and the addressing problem in data communication systems they also proved that the determinant of the distance matrix of a tree is a function of the number of vertices only since then several mathematicians were interested in studying the spectral properties of the distance matrix of a connected graph computing the distance characteristic polynomial and its coefficients was the first research subject of interest thereafter the eigenvalues attracted much more attention in the present paper we report on the results related to the distance matrix of a graph and its spectral properties

complete multipartite graphs are determined by their distance spectra abstract in this paper we prove that the complete multipartite graphs are determined by their distance spectra which confirms the conjecture proposed by lin hong wang and shu 2013 7 although it is well known that the complete multipartite graphs cannot be determined by their adjacency spectra

adaptive relaxation labeling current implementation of probabilistic relaxation labeling prl is based on stationary compatibility coefficients scc s such labeling frequently diverges from an achieved minimum labeling error in this correspondence it is shown that by having nonstationary compatibility coefficients nscc s the prl stabilizes about the minimum error which is obtained during the early iterations also a noniterative labeling algorithm which uses nscc and has a performance similar to that of the modified prl is suggested

confidence and conflict in multivariate calibration summary multivariate calibration involves using an estimated relationship between a multivariate response y and an explanatory vector x to predict unknown x in future from further observed responses in controlled calibration with multivariate normal errors the profile likelihood function for the unknown x denoted 4 displays a term which measures the mutual inconsistency of the given response vector denoted z in predicting 4 this inconsistency diagnostic fundamentally differentiates the behaviour of likelihood based and bayes confidence regions from those of the unconditional sampling approach in addition the diagnostic serves to pinpoint an inadequate response vector z

triplet markov fields for the classification of complex structure data we address the issue of classifying complex data we focus on three main sources of complexity namely the high dimensionality of the observed data the dependencies between these observations and the general nature of the noise model underlying their distribution we investigate the recent triplet markov fields and propose new models in this class designed for such data and in particular allowing very general noise models in addition our models can handle the inclusion of a learning step in a consistent way so that they can be used in a supervised framework one advantage of our models is that whatever the initial complexity of the noise model parameter estimation can be carried out using state of the art bayesian clustering techniques under the usual simplifying assumptions as generative models they can be seen as an alternative in the supervised case to discriminative conditional random fields identifiability issues underlying the models in the nonsupervised case are discussed while the models performance is illustrated on simulated and real data exhibiting the mentioned various sources of complexity

comments on application of the conditional population mixture model to image segmentation in the above correspondence1 a maximum likelihood method is proposed for estimating class memberships and underlying statistical parameters within the context of distribution mixtures in the present comment it is pointed out that biases are incurred in parameter estimation that the class memberships and parameters are conceptually different and therefore that the so called standard mixture likelihood is to be preferred also in the correspondence 1 akaike s information criterion aic is used to choose the number of classes in the mixture here a brief theoretical caveat is issued

sequential and parallel image restoration neural network implementations sequential and parallel image restoration algorithms and their implementations on neural networks are proposed for images degraded by linear blur and contaminated by additive white gaussian noise maximum a posteriori map estimation and regularization theory lead to the same high dimension convex optimization problem the commonly adopted strategy in using neural networks for image restoration is to map the objective function of the optimization problem into the energy of a predefined network taking advantage of its energy minimization properties departing from this approach we propose neural implementations of iterative minimization algorithms which are first proved to converge the developed schemes are based on modified hopfield 1985 networks of graded elements with both sequential and parallel updating schedules an algorithm supported on a fully standard hopfield network binary elements and zero autoconnections is also considered robustness with respect to finite numerical precision is studied and examples with real images are presented

maximum likelihood reconstruction for emission tomography previous models for emission tomography et do not distinguish the physics of et from that of transmission tomography we give a more accurate general mathematical model for et where an unknown emission density x y z generates and is to be reconstructed from the number of counts n d in each of d detector units d within the model we give an algorithm for determining an estimate of which maximizes the probability p n of observing the actual detector count data n over all possible densities let independent poisson variables n b with unknown means b b 1 b represent the number of unobserved emissions in each of b boxes pixels partitioning an object containing an emitter suppose each emission in box b is detected in detector unit d with probability p b d d 1 d with p b d a one step transition matrix assumed known we observe the total number n n d of emissions in each detector unit d and want to estimate the unknown b b 1 b for each the observed data n has probability or likelihood p n the em algorithm of mathematical statistics starts with an initial estimate 0 and gives the following simple iterative procedure for obtaining a new estimate new from an old estimate old to obtain k k 1 2 new b old b dd 1 n d p b d old b p b d b 1 b

cospectrality of graphs with respect to distance matrices abstract the distance distance laplacian and distance signless laplacian spectra of a connected graph g are the spectra of the distance distance laplacian and distance signless laplacian matrices of g two graphs are said to be cospectral with respect to the distance resp distance laplacian or distance signless laplacian matrix if they share the same distance resp distance laplacian or distance signless laplacian spectrum if a graph g does not share its spectrum with any other graph we say g is determined by its spectrum in this paper we are interested in the cospectrality with respect to the three distance matrices first we report on a numerical study in which we looked into the spectra of the distance distance laplacian and distance signless laplacian matrices of all the connected graphs on up to 10 vertices then we prove some theoretical results about what we can deduce about a graph from these spectra among other results we identify some of the graphs determined by their distance laplacian or distance signless laplacian spectra

regularization theory in image restoration the stabilizing functional approach several aspects of the application of regularization theory in image restoration are presented this is accomplished by extending the applicability of the stabilizing functional approach to 2 d ill posed inverse problems inverse restoration is formulated as the constrained minimization of a stabilizing functional the choice of a particular quadratic functional to be minimized is related to the a priori knowledge regarding the original object through a formulation of image restoration as a maximum a posteriori estimation problem this formulation is based on image representation by certain stochastic partial differential equation image models the analytical study and computational treatment of the resulting optimization problem are subsequently presented as a result a variety of regularizing filters and iterative regularizing algorithms are proposed a relationship between the regularized solutions proposed and optimal wiener estimation is identified the filters and algorithms proposed are evaluated through several experimental results

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

exact output tracking theory for systems with parameter jumps in this paper we consider the exact output tracking problem for systems with parameter jumps necessary and sufficient conditions are derived for the elimination of switching introduced output transient previous works have studied this problem by developing a regulator that maintains exact tracking through parameter jumps switches such techniques are however only applicable to minimum phase systems in contrast our approach is applicable to nonminimum phase systems and obtains bounded but possibly non causal solutions if the reference trajectories are generated by an exo system then we develop an exact tracking controller in a feedback form as in standard regulator theory we obtain a linear map from the states of the exo system to the desired system state which is defined via a matrix differential equation the constant solution of this differential equation provides asymptotic tracking and coincides with the feedback law used in standard regulator theory the obtained results are applied to a simple flexible manipulator with jumps in the pay load mass

bayes smoothing algorithms for segmentation of binary images modeled by markov random fields a new image segmentation algorithm is presented based on recursive bayes smoothing of images modeled by markov random fields and corrupted by independent additive noise the bayes smoothing algorithm yields the a posteriori distribution of the scene value at each pixel given the total noisy image in a recursive way the a posteriori distribution together with a criterion of optimality then determine a bayes estimate of the scene the algorithm presented is an extension of a 1 d bayes smoothing algorithm to 2 d and it gives the optimum bayes estimate for the scene value at each pixel computational concerns in 2 d however necessitate certain simplifying assumptions on the model and approximations on the implementation of the algorithm in particular the scene noiseless image is modeled as a markov mesh random field a special class of markov random fields and the bayes smoothing algorithm is applied on overlapping strips horizontal vertical of the image consisting of several rows columns it is assumed that the signal scene values vector sequence along the strip is a vector markov chain since signal correlation in one of the dimensions is not fully used along the edges of the strip estimates are generated only along the middle sections of the strips the overlapping strips are chosen such that the union of the middle sections of the strips gives the whole image the bayes smoothing algorithm presented here is valid for scene random fields consisting of multilevel discrete or continuous random variables

distance spectral radius of trees with given number of segments abstract a segment of a tree is a path subtree whose terminal vertices are branching or pendant vertices we determine the structure of trees that maximize the distance spectral radius among trees with given order and number of segments and among trees with given order and number of vertices of degree 2 respectively

on the foundations of relaxation labeling processes a large class of problems can be formulated in terms of the assignment of labels to objects frequently processes are needed which reduce ambiguity and noise and select the best label among several possible choices relaxation labeling processes are just such a class of algorithms they are based on the parallel use of local constraints between labels this paper develops a theory to characterize the goal of relaxation labeling the theory is founded on a definition of con sistency in labelings extending the notion of constraint satisfaction in certain restricted circumstances an explicit functional exists that can be maximized to guide the search for consistent labelings this functional is used to derive a new relaxation labeling operator when the restrictions are not satisfied the theory relies on variational cal culus it is shown that the problem of finding consistent labelings is equivalent to solving a variational inequality a procedure nearly identical to the relaxation operator derived under restricted circum stances serves in the more general setting further a local convergence result is established for this operator the standard relaxation labeling formulas are shown to approximate our new operator which leads us to conjecture that successful applications of the standard methods are explainable by the theory developed here observations about con vergence and generalizations to higher order compatibility relations are described

on the distance laplacian spectra of graphs abstract the distance laplacian matrix of a connected graph g is defined in 2 3 and it is proved that for a graph g on n vertices if the complement of g is connected then the second smallest distance laplacian eigenvalue is strictly greater than n in this article we consider the graphs whose complement is a tree or a unicyclic graph and characterize the graphs among them having n 1 as the second smallest distance laplacian eigenvalue we prove that the largest distance laplacian eigenvalue of a path is simple and the corresponding eigenvector has the similar property like that of a fiedler vector

on the statistical analysis of dirty pictures may 7th 1986 professor a f m smith in the chair summary a continuous two dimensional region is partitioned into a fine rectangular array of sites or pixels each pixel having a particular colour belonging to a prescribed finite set the true colouring of the region is unknown but associated with each pixel there is a possibly multivariate record which conveys imperfect information about its colour according to a known statistical model the aim is to reconstruct the true scene with the additional knowledge that pixels close together tend to have the same or similar colours in this paper it is assumed that the local characteristics of the true scene can be represented by a nondegenerate markov random field such information can be combined with the records by bayes theorem and the true scene can be estimated according to standard criteria however the computational burden is enormous and the reconstruction may reflect undesirable largescale properties of the random field thus a simple iterative method of reconstruction is proposed which does not depend on these large scale characteristics the method is illustrated by computer simulations in which the original scene is not directly related to the assumed random field some complications including parameter estimation are discussed potential applications are mentioned briefly

markov random field texture models we consider a texture to be a stochastic possibly periodic two dimensional image field a texture model is a mathematical procedure capable of producing and describing a textured image we explore the use of markov random fields as texture models the binomial model where each point in the texture has a binomial distribution with parameter controlled by its neighbors and number of tries equal to the number of gray levels was taken to be the basic model for the analysis a method of generating samples from the binomial model is given followed by a theoretical and practical analysis of the method s convergence examples show how the parameters of the markov random field control the strength and direction of the clustering in the image the power of the binomial model to produce blurry sharp line like and blob like textures is demonstrated natural texture samples were digitized and their parameters were estimated under the markov random field model a hypothesis test was used for an objective assessment of goodness of fit under the markov random field model overall microtextures fit the model well the estimated parameters of the natural textures were used as input to the generation procedure the synthetic microtextures closely resembled their real counterparts while the regular and inhomogeneous textures did not

brief paper stability results for linear parameter varying and switching systems in this paper stabilizability problems for discrete and continuous time linear systems with switching or polytopic uncertainties are dealt with both state feedback and full information controllers are considered possibly with an integral action several cases will be analyzed depending on the control the uncertainty and the presence of uncertainties in the input matrix a complete overview of the stabilizability implications among these different cases will be given

maximum likelihood discriminant analysis on the plane using a markovian model of spatial context abstract a method is presented for utilising spatial information in performing discriminant analysis on multivariate data at each point on a regular lattice as for example with landsat the spatial information is modelled as a particularly simple markov random field the discriminant analysis is based on the maximum likelihood method the results are compared with alternatives by monte carlo simulation and seem to be encouraging

unsupervised image restoration and edge location using compound gauss markov random fields and the mdl principle discontinuity preserving bayesian image restoration typically involves two markov random fields one representing the image intensities gray levels to be recovered and another one signaling discontinuities edges to be preserved the usual strategy is to perform joint maximum a posterori map estimation of the image and its edges which requires the specification of priors for both fields instead of taking an edge prior we interpret discontinuities in fact their locations as deterministic unknown parameters of the compound gauss markov random field cgmrf which is assumed to model the intensities this strategy should allow inferring the discontinuity locations directly from the image with no further assumptions however an additional problem emerges the number of parameters edges is unknown to deal with it we invoke the minimum description length mdl principle according to mdl the best edge configuration is the one that allows the shortest description of the image and its edges taking the other model parameters noise and cgmrf variances also as unknown we propose a new unsupervised discontinuity preserving image restoration criterion implementation is carried out by a continuation type iterative algorithm which provides estimates of the number of discontinuities their locations the noise variance the original image variance and the original image itself restored image experimental results with real and synthetic images are reported

trajectory tracking control of bimodal piecewise affine systems this paper deals with a trajectory tracking problem for a class of bimodal piecewise affine systems which is inherently difficult because of the discontinuous changes of their vector fields first we introduce an error variable and an error system as a generalization of the tracking error and its system as an error variable a function switched by the mode of a piecewise affine system is adopted to overcome the inherent difficulty in trajectory tracking control of piecewise affine systems next we design a tracking controller which stabilizes the error system using a lyapunov like function which can be applied to systems including state jumps furthermore the feasibility condition of tracking for siso piecewise linear systems is simplified finally a numerical example is given to illustrate the effectiveness of the proposed method

regulator synthesis for bimodal linear systems this technical note presents a regulator synthesis approach for switched bimodal linear systems where it is desired to achieve regulation against known disturbance or reference signals first a set of observer based q parameterized stabilizing controllers for the switched system is constructed then regulation conditions for the switched system are obtained and a regulator synthesis method is derived based on iteratively solving a set of linear matrix inequalities finally a numerical example is presented to illustrate the effectiveness of the proposed method

on generalized distance energy of graphs abstract for the distance matrix d g and diagonal matrix of the vertex transmissions t r g of a simple connected graph g the generalized distance matrix d g is the convex combinations of t r g and d g and is defined as d g t r g 1 d g for 0 1 if 1 2 n are the eigenvalues of d g we define the generalized distance energy of the graph g as e d g i 1 n i 2 w g n where w g is the wiener index of g this is analogous to the energies associated with the distance laplacian and distance signless laplacian matrices of g we obtain upper and lower bounds for the generalized distance energy of graphs in terms of various parameters associated with the structure of the graph g we show that for 1 2 1 the complete bipartite graph has the minimum generalized distance energy among all connected bipartite graphs and for 0 2 n 3 n 2 the star graph has the minimum generalized distance energy among all trees

interval estimation in discrimination the multivariate normal equal covariance case summary a general approach to discrimination problems is described which emphasizes the viewpoint of individual cases and expresses the uncertainty with which individual decisions are made it is applied to the particular case of two multivariate normal distributions with a common covariance matrix an affine transformation to which the key quantities are invariant simplifies discussion of the distribution of the estimated log odds ratio and suggests a practically useful plot which generalizes the usual display of discriminant scores and has several attractive features an illustration is given two approximate interval estimation techniques for the true log odds ratio are described and evaluated in a range of practical situations by a simulation study some further applications of our general approach are briefly indicated

on the covariance of two noncentral f random variables and the variance of the estimated liner discriminant function summary the covariance of two noncentral f random variables with a common covariance matrix is obtained from which the variance of their difference follows easily an application of this latter result in discriminant analysis is then given

on the distance spectrum of graphs abstract let g be a connected graph and d g be the distance matrix of g suppose that 1 d 2 d n d are the d eigenvalues of g in this paper we characterize all connected graphs with n d 2 furthermore we characterize all connected graphs of diameter 2 with exactly three d eigenvalues when 1 d is not an integer we also conjecture that the complete k partite graph is determined by its d spectrum

analysis of field experiments by least squares smoothing summary assuming a smooth trend plus independent error model for the environmental effects in the yields of a field plot experiment least squares smoothing methods are developed to estimate both the treatment effects and the unknown trend treatment estimates are closely related to those resulting from a generalized least squares analysis in which the covariance structure for the environmental effects has a particular form however the main emphases are on the accuracy of treatment estimates under a fixed smooth trend plus error model and the exploratory power of the basic method to isolate trend effects of unknown form although the detailed development is for the one dimensional case generalizations of the smoothness concept and extensions to two dimensions are also discussed application of the basic method is illustrated on three data sets and the results compared with other analyses

observer based control of piecewise affine systems this paper presents a new synthesis method for both state and dynamic output feedback control of a class of hybrid systems called piecewise affine pwa systems the synthesis procedure delivers stabilizing controllers that can be proven to give either asymptotic or exponential convergence rates the synthesis method builds on existing pwa stability analysis tools by transforming the design into a closed loop analysis problem wherein the controller parameters are unknown more specifically the proposed technique formulates the search for a piecewise quadratic control lyapunov function and a piecewise affine control law as an optimization problem subject to linear constraints and a bilinear matrix inequality the linear constraints in the synthesis guarantee that sliding modes are not generated at the switching the resulting optimization problem is known to be hard but suboptimal solutions can be obtained using the three iterative algorithms presented in the paper the new synthesis technique allows con

parallel and deterministic algorithms from mrfs surface reconstruction deterministic approximations to markov random field mrf models are derived one of the models is shown to give in a natural way the graduated nonconvexity gnc algorithm proposed by a blake and a zisserman 1987 this model can be applied to smooth a field preserving its discontinuities a class of more complex models is then proposed in order to deal with a variety of vision problems all the theoretical results are obtained in the framework of statistical mechanics and mean field techniques a parallel iterative algorithm to solve the deterministic equations of the two models is presented together with some experiments on synthetic and real images

bayesian estimation of ventricular contours in angiographic images a method for left ventricular contour determination in digital angiographic images is presented the problem is formulated in a bayesian framework adopting as the estimation criterion the maximum a posterior probability map the true contour is modeled as a one dimensional noncausal gauss markov random field and the observed image is described as the superposition of an ideal image deterministic function of the real contour with white gaussian noise the proposed algorithm estimates simultaneously the contour and the model parameters by implementing an adaptive version of the iterated conditional modes algorithm the convergence of this scheme is proved and its performance evaluated on both synthetic and real angiographic images the method exhibits robustness against image artifacts and the contours obtained are considered good by expert clinicians being completely data driven and fast the proposed algorithm is suitable for routine clinical use

cospectral graphs and the generalized adjacency matrix abstract let j be the all ones matrix and let a denote the adjacency matrix of a graph an old result of johnson and newman states that if two graphs are cospectral with respect to yj a for two distinct values of y then they are cospectral for all y here we will focus on graphs cospectral with respect to yj a for exactly one value y of y we call such graphs y cospectral it follows that y is a rational number and we prove existence of a pair of y cospectral graphs for every rational y in addition we generate by computer all y cospectral pairs on at most nine vertices recently chesnokov and the second author constructed pairs of y cospectral graphs for all rational y 0 1 where one graph is regular and the other one is not this phenomenon is only possible for the mentioned values of y and by computer we find all such pairs of y cospectral graphs on at most eleven vertices

segmentation based on fusion of range and intensity images using robust trimmed methods abstract this paper proposes a segmentation algorithm based on fusion of range and intensity images using robust trimmed methods based on the bayesian theory a priori knowledge is represented using the markov random field mrf a maximum a posteriori map estimator is constructed using the edge features extracted from both range and intensity images objects are represented by a number of local planar surfaces in range images and the parametric space for surface representation is constructed with the surface parameters estimated pixel by pixel based on the least trimmed squares lts method whereas in intensity images the trimmed variance is adopted as the feature for edge extraction a final edge map is obtained by the map estimator that is constructed using the likelihood functions based on the edge information obtained from range and intensity images finally an image is segmented using the fused edge map computer simulation results show that our new segmentation algorithm effectively segments test images independent of shadow noise and lighting environment

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

recursive structure of noncausal gauss markov random fields an approach is developed for noncausal gauss markov random fields gmrfs that enables the use of recursive procedures while retaining the noncausality of the field recursive representations are established that are equivalent to the original field this is achieved by first presenting a canonical representation for gmrfs that is based on the inverse of the covariance matrix which is called the potential matrix it is this matrix rather than the field covariance that reflects in a natural way the mrf structure from its properties two equivalent one sided representations are derived each of which is obtained as the successive iterates of a riccati type equation for homogeneous fields these unilateral descriptions are symmetrized versions of each other the study of only one riccati equation being required it is proven that this riccati equation converges at a geometric rate therefore the one sided representations are asymptotically invariant these unilateral representations make it possible to process the fields with well known recursive techniques such as kalman bucy filters and two point smoothers

a statistical model for positron emission tomography abstract positron emission tomography pet still in its research stages is a technique that promises to open new medical frontiers by enabling physicians to study the metabolic activity of the body in a pictorial manner much as in x ray transmission tomography and other modes of computerized tomography the quality of the reconstructed image in pet is very sensitive to the mathematical algorithm to be used for reconstruction in this article we tailor a mathematical model to the physics of positron emissions and we use the model to describe the basic image reconstruction problem of pet as a standard problem in statistical estimation from incomplete data we describe various estimation procedures such as the maximum likelihood ml method using the em algorithm the method of moments and the least squares method a computer simulation of a pet experiment is then used to demonstrate the ml and the least squares reconstructions the main purposes of this article are to report on what we believe is an

graphical and recursive models for contingency tables summary we discuss two classes of models for contingency tables graphical and recursive models both of which arise from restrictions that are expressible as conditional independencies of variable pairs the first of these is a subclass of hierarchical log linear models each of its models can be represented by an undirected graph in the second class each model corresponds to a particular kind of a directed graph instead and can be characterized by a nontrivial factorization of the joint distribution in terms of response variables we derive decomposable or multiplicative models as the intersecting class this result has useful consequences for exploratory types of analysis as well as for the model interpretation we can give an aid for detecting well fitting decomposable models in a transformation of the observed contingency table and each decomposable model may be interpreted with the help of an undirected or directed graph

on the second largest eigenvalue of the generalized distance matrix of graphs abstract for a simple connected graph g let d g t r g d l g and d q g respectively be the distance matrix the diagonal matrix of the vertex transmissions distance laplacian matrix and the distance signless laplacian matrix of a graph g the generalized distance matrix d g is a convex linear combinations of t r g and d g and defined as d g t r g 1 d g 0 1 as d 0 g d g 2 d 1 2 g d q g d 1 g t r g and d g d g d l g this matrix reduces to merging the distance spectral distance laplacian spectral and distance signless laplacian spectral theories in this paper we take effort to obtain some upper and lower bounds for the second largest eigenvalue of the generalized distance matrix of graphs in terms of various graph parameters the graphs attaining the corresponding bounds are characterized as application we give a confirmation to a conjecture about the second largest distance signless laplacian eigenvalue of a connected graph due to aouchiche and hansen 6 we also show that the star graph s n has the smallest second largest generalized distance eigenvalue among all trees of order n as application we give a confirmation to a conjecture about the second largest distance signless laplacian eigenvalue of a tree due to aouchiche and hansen 10