on some properties of graph irregularity indices with a particular regard to the σ index abstract some well known graph irregularity indices of a connected graph g are investigated our study is focused mainly on the comparison of the bell s degree variance var g and the collatz sinogowitz irregularity index cs g with the recently introduced g irregularity index it is a degree based topological invariant calculated as g f g 2 m 2 g where m2 g is the second zagreb index f g d 3 v and d v is the degree of the vertex v in g by introducing the notion of the complete split like graphs representing a broad subclass of bidegreed connected graphs it is shown that for these graphs the equality g n 2 v a r g holds

mimicking counterfactual outcomes to estimate causal effects in observational studies treatment may be adapted to covariates at several times without a fixed protocol in continuous time treatment influences covariates which influence treatment which influences covariates and so on then even time dependent cox models cannot be used to estimate the net treatment effect structural nested models have been applied in this setting structural nested models are based on counterfactuals the outcome a person would have had had treatment been withheld after a certain time previous work on continuous time structural nested models assumes that counterfactuals depend deterministically on observed data while conjecturing that this assumption can be relaxed this article proves that one can mimic counterfactuals by constructing random variables solutions to a differential equation that have the same distribution as the counterfactuals even given past observed data these mimicking variables can be used to estimate the parameters of structural nested models without assuming the treatment effect to be deterministic

the walk partition and colorations of a graph we study the partitions of a graph g with vertices labeled 1 2 n a matrix notation for partitions is devised which reflects a partial order among partitions and provides a matrix characterization in terms of the adjacency matrix a of g of a coloration or equitable partition an especially useful partition is one based on the number of walks issuing from each vertex this is not generally a coloration sufficient conditions are given but nevertheless plays a special role relative to colorations the spectrum of g sp g has a main part the set of eigenvalues with an eigenvector not orthogonal to e the column of 1s we disprove a conjecture of harary and schwenk about the main part of the spectrum and prove an alternate characterization the walk partition appears here and also in a relation between the main parts of the spectra of g and its complement

causal inference for complex longitudinal data the continuous case we extend robins theory of causal inference for complex longitudinal data to the case of continuously varying as opposed to discrete covariates and treatments in particular we establish versions of the key results of the discrete theory the g computation formula and a collection of powerful characterizations of the g null hypothesis of no treatment eect this is accomplished under natural continuity hypotheses concerning the conditional distributions of the outcome variable and of the covariates given the past we also show that our assumptions concerning counterfactual variables place no restriction on the joint distribution of the observed variables thus in a precise sense these assumptions are for free or if you prefer harmless

star complements and exceptional graphs let g be a finite graph of order n with an eigenvalue of multiplicity k thus the eigenspace of a 0 1 adjacency matrix of g has dimension k astar complement for in g is an induced subgraph g x of g such that x k and g x does not have as an eigenvalue an exceptional graph is a connected graph other than a generalized line graph whose eigenvalues lie in 2 we establish some properties of star complements and of eigenvectors of exceptional graphs with least eigenvalue 2

a short remark on graphs with two main eigenvalues abstract we give a short proof of a conjecture on graphs with two main eigenvalues posed by reti in applied mathematics and computation 344 345 2019 107 115

a robust optimal model matching control a deductive design approach is proposed to develop an optimal feedback control while preserving quantitative robustness and noise rejection properties the design procedure involves the minimization of a suitably selected performance index to reflect a required model matching following objective gradient projection schemes applied to the performance index and feasible constraints are used to accomplish the design tradeoff a mechanical manipulator control system example is employed as a vehicle to illustrate the overall design selection and optimization uncertainty of the system description is included as a feature of the example

some results on graph spectra abstract this paper presents a variety of results on graph spectra the number of main eigenvalues of a graph is shown to be equal to the rank of an associated matrix we establish a condition for a graph to have exactly two main eigenvalues and then show how to evaluate them and their associated eigenvectors it is shown that the main eigenvalues and corresponding eigenvectors of a graph determine those of its complement we generalize to any eigenvalue a condition for 0 and 1 to be eigenvalues of a graph and its complement respectively finally we generalize to non simple eigenvalues a result on the components of an eigenvector associated with a simple eigenvalue

note on the generalized spectral characterization of graphs having an isolated vertex abstract in this paper some sufficient conditions are given on a graph g under which it is proved that g x is determined by the generalized spectrum iff g is determined by the generalized spectrum where g x is the graph obtained from the graph g by adding an isolated vertex x

main eigenvalues and κ τ regular sets abstract a regular set is a subset of the vertices of a graph g inducing a regular subgraph such that every vertex not in the subset has neighbors in it a main eigenvalue of the adjacency matrix a of a graph g has an eigenvector not orthogonal to the all one vector j for graphs with a regular set a necessary and sufficient condition for an eigenvalue be non main is deduced and the main eigenvalues are characterized these results are applied to the construction of infinite families of bidegreed graphs with two main eigenvalues and the same spectral radius index and some relations with strongly regular graphs are obtained finally the determination of regular sets is analyzed

a bound on the spectral radius of graphs in terms of their zagreb indices abstract the first and the second zagreb index of a graph usually defined as the sum of the squares of degrees over all vertices and the sum of the products of degrees of edge endvertices over all edges respectively are tightly related to the numbers of walks of length two and three in the graph we provide here a lower bound on the spectral radius of adjacency matrix a of graph in terms of its zagreb indices based on the properties of the least square approximation of the vector a 2 j with the vectors aj and j where j is the all one vector the bound is sharp for all graphs with two main eigenvalues surpassing the range of sharpness of other bounds among connected graphs

a note on hoffman type identities of graphs an eigenvalue of a graph g is called main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero hoffman a j hoffman on the polynomial of a graph amer math monthly 70 1963 30 36 proved that g is a connected k regular graph if and only if n i 2t a ii i 2t k i j where i is the unit matrix and j the all one matrix and 1 k 2 t are all distinct eigenvalues of adjacency matrix a g in this note some generalizations of hoffman identity are presented by means of main eigenvalues

the integral graphs with index 3 and exactly two main eigenvalues a graph is called integral if the spectrum of its adjacency matrix has only integral eigenvalues an eigenvalue of a graph is called main eigenvalue if it has an eigenvector such that the sum of whose entries is not equal to zero in this paper we show that there are exactly 25 connected integral graphs with exactly two main eigenvalues and index 3

eigenvalues assignment by a particular lyapunov type equation this paper proposes a particular lyapunov type equation and uses its solution to synthesize a state feedback control such that the eigenvalues of the closed loop multiple inputs system can be assigned to the left half plane with arbitrary stability degree

robust pole assignment in linear state feedback numerical methods are described for determining robust or well conditioned solutions to the problem of pole assignment by state feedback the solutions obtained are such that the sensitivity of the assigned poles to perturbations in the system and gain matrices is minimized it is shown that for these solutions upper bounds on the norm of the feedback matrix and on the transient response are also minimized and a lower bound on the stability margin is maximized a measure is derived which indicates the optimal conditioning that may be expected for a particular system with a given set of closed loop poles and hence the suitability of the given poles for assignment

construction of graphs with exactly k main eigenvalues abstract given a simple graph g the vertex partition v g v 1 v 2 v r is said to be an equitable partition if for any u v i v j n g u b i j is a constant whenever 1 i j r an equitable partition leads to a divisor g of g which is the directed multigraph with vertices v 1 v 2 v r and b i j arcs from v i to v j conversely a directed multigraph may not be a divisor of some simple graph in this paper we give a necessary and sufficient condition for a directed multigraph to be the divisor of some simple graph by the way we give a method to construct many classes of connected graphs with exactly k main eigenvalues for any positive integer k

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

a note on graphs with exactly two main eigenvalues abstract in this note we consider connected graphs with exactly two main eigenvalues we will give several constructions for them and as a consequence we show the existence of a family of such graphs with an unbounded number of distinct valencies

statistical modeling of causal effects in continuous time this article studies the estimation of the causal effect of a time varying treatment on time to an event or on some other continuously distributed outcome the paper applies to the situation where treatment is repeatedly adapted to time dependent patient characteristics the treatment effect cannot be estimated by simply conditioning on these time dependent patient characteristics as they may themselves be indications of the treatment effect this time dependent confounding is common in observational studies robins 1992 biometrika 79 321 334 1998b encyclopedia of biostatistics 6 4372 4389 has proposed the so called structural nested models to estimate treatment effects in the presence of time dependent confounding in this article we provide a conceptual framework and formalization for structural nested models in continuous time we show that the resulting estimators are consistent and asymptotically normal moreover as conjectured in robins 1998b encyclopedia of biostatistics 6 4372 4389 a test for whether treatment affects the outcome of interest can be performed without specifying a model for treatment effect we illustrate the ideas in this article with an example

statistics and causal inference abstract problems involving causal inference have dogged at the heels of statistics since its earliest days correlation does not imply causation and yet causal conclusions drawn from a carefully designed experiment are often valid what can a statistical model say about causation this question is addressed by using a particular model for causal inference holland and rubin 1983 rubin 1974 to critique the discussions of other writers on causation and causal inference these include selected philosophers medical researchers statisticians econometricians and proponents of causal modeling

semiparametric estimation of structural failure time models in continuous time processes structural failure time models are causal models for estimating the effect of time varying treatments on a survival outcome g estimation and artificial censoring have been proposed for estimating the model parameters in the presence of time dependent confounding and administrative censoring however most existing methods require manually pre processing data into regularly spaced data which may invalidate the subsequent causal analysis moreover the computation and inference are challenging due to the nonsmoothness of artificial censoring we propose a class of continuous time structural failure time models that respects the continuous time nature of the underlying data processes under a martingale condition of no unmeasured confounding we show that the model parameters are identifiable from a potentially infinite number of estimating equations using the semiparametric efficiency theory we derive the first semiparametric doubly robust estimators which are consistent if the model for the treatment process or the failure time model but not necessarily both is correctly specified moreover we propose using inverse probability of censoring weighting to deal with dependent censoring in contrast to artificial censoring our weighting strategy does not introduce nonsmoothness in estimation and ensures that resampling methods can be used for inference

spectra signless laplacian and laplacian spectra of complementary prisms of graphs abstract the complementary prism g g of a graph g is obtained from the disjoint union of g and its complement g by adding an edge for each pair of vertices v v where v is in g and its copy v is in g the petersen graph c 5 c 5 and for n 2 the corona product of k n and k 1 which is k n k n are examples of complementary prisms this paper is devoted to the computation of eigenpairs of the adjacency signless laplacian and laplacian matrices of a complementary prism g g in terms of the eigenpairs of the corresponding matrices of g particular attention is given to the complementary prisms of regular graphs furthermore petersen graph is shown to be the unique complementary prism which is a strongly regular graph

estimation of the time dependent accelerated failure time model in the presence of confounding factors summary cox oakes 1984 p 66 introduced the strong version of the accelerated failure time model with time dependent exposures we provide conditions under which this model could be used to estimate from observational data the causal effect of a time varying exposure or treatment on time to an event of interest in the presence of time dependent confounding variables we propose a class of semiparametric tests and estimators for the model parameters this class contains an estimator that is semipara metric efficient in the sense of begun et al 1983 the purpose of this paper is first to provide conditions under which it would be possible to estimate from observational data the causal effect of a time dependent treatment or exposure on time to an event of interest in the presence of time dependent confounding covariates and then to construct estimators for the treatment effect under these condi tions our approach will be based on the semiparametric estimation of the parameters of the strong version of the accelerated failure time model with time dependent exposures or treatments introduced by cox oakes 1984 p 66 the usual approach to the estimation of the effect of a time varying treatment on survival is to model the hazard of failure at t as a function of past treatment history using a time dependent proportional hazards model we show that the usual approach may be biased whether or not one further adjusts for past confounder history in the analysis when a there exists a time dependent risk factor for or predictor of the event of interest that also predicts subsequent treatment and b past treatment history predicts subsequent risk factor level the following four examples demonstrate conditions a and b will often be true in an observational study in which there is treatment by indication the drug zidovudine formerly azt used in the treatment of the acquired immunodeficiency syndrome is a direct red blood cell toxin that is contra indicated in anaemic subjects i e subjects with depressed red cell counts since the toxic effects of zidovudine can worsen the anaemia further anaemic patients are at increased risk of death thus in an observational study of the effect of zidovudine on survival of patients with the acquired immunodeficiency syndrome anaemia is both a risk factor for death