comment on the problem of deficiency indices for discrete schrodinger operators on locally finite graphs j math phys 52 063512 2011 in this comment we answer negatively to our conjecture concerning the deficiency indices more precisely given any non negative integer n there is locally finite graph on which the adjacency matrix has deficiency indices n n

examination of the 1 d expansion method from exact enumeration for a self interacting self avoiding walk the 1 d expansion method for polymer chains is examined by comparing these expansions for several thermodynamic and structural quantities with the results of standard series analysis of exact enumeration data the comparisons cover a wide range of spatial dimensions d including non integer ones and are performed for particular values of interaction energy good agreement is generally found for d 4 whereas discrepancies become conspicuous as d decreases to d 2 reasonable values are obtained for the exponents v and gamma in d 2 4 by applying the coherent anomaly method of suzuki 1986 to our 1 d expansions through fifth order in d 1

statistics of collapsing lattice animals we consider several lattice models of branched polymers in dilute solution in which the polymer molecule is modelled as a tree or animal in general the thermodynamic properties of the polymer are determined by monomer monomer and monomer solvent interactions we examine a two variable model which incorporates both types of interaction and discuss its relationship to other models which have previously been investigated in particular we discuss the collapse transition in these models

information processing in complex networks graph entropy and information functionals this paper introduces a general framework for defining the entropy of a graph our definition is based on a local information graph and on information functionals derived from the topological structure of a given graph more precisely an information functional quantifies structural information of a graph based on a derived probability distribution such a probability distribution leads directly to an entropy of a graph then the structural information content of a graph will be is interpreted and defined as the derived graph entropy another major contribution of this paper is the investigation of relationships between graph entropies in addition to this we provide numerical results demonstrating not only the feasibility of our method which has polynomial time complexity but also its usefulness with regard to practical applications aiming to an understanding of information processing in complex networks

subgraph centrality and walk regularity abstract matrix based centrality measures have enjoyed significant popularity in network analysis in no small part due to our ability to rigorously analyze their behavior as parameters vary recent work has considered the relationship between subgraph centrality which is defined using the matrix exponential f x exp x and the walk structure of a network in a walk regular graph the number of closed walks of each length must be the same for all nodes implying uniform f subgraph centralities for any f or maximum f walk entropy we consider when non walk regular graphs can achieve maximum entropy calling such graphs entropic for parameterized measures we are also interested in which values of the parameter witness this uniformity to date only one entropic graph has been identified with only two witnessing parameter values raising the question of how many such graphs and parameters exist we resolve these questions by constructing infinite families of entropic graphs as well as a family of witnessing parameters with a limit point at zero

relationships between growth constants for animals and trees lattice theory we consider the growth constants of several types of animals on d dimensional hypercubic lattices a combination of rigorous inequalities and 1 d expansions leads us to conjecture a strict ordering of these growth constants these results are useful in the analysis of models of branched polymer behaviour

bose hubbard model on two dimensional line graphs we construct a basis for the many particle ground states of the positive hopping bose hubbard model on line graphs of finite 2 connected planar bipartite graphs at sufficiently low filling factors the particles in these states are localized on non intersecting vertex disjoint cycles of the line graph which correspond to non intersecting edge disjoint cycles of the original graph the construction works up to a critical filling factor at which the cycles are close packed

efficiency comparisons of rank and permutation tests based on summary statistics computed from repeated measures data a popular method of using repeated measures data to compare treatment groups in a clinical trial is to summarize each individual s outcomes with a scalar summary statistic and then to perform a two group comparison of the resulting statistics using a rank or permutation test many different types of summary statistics are used in practice including discrete and continuous functions of the underlying repeated measures data when the repeated measures processes of the comparison groups differ by a location shift at each time point the asymptotic relative efficiency of continuous summary statistics that are linear functions of the repeated measures has been determined and used to compare tests in this class however little is known about the non null behaviour of discrete summary statistics about continuous summary statistics when the groups differ in more complex ways than location shifts or where the summary statistics are not linear functions of the repeated measures indeed even simple distributional structures on the repeated measures variables can lead to complex differences between the distribution of common summary statistics of the comparison groups the presence of left censoring of the repeated measures which can arise when these are laboratory markers with lower limits of detection further complicates the distribution of and hence the ability to compare summary statistics this paper uses recent theoretical results for the non null behaviour of rank and permutation tests to examine the asymptotic relative efficiencies of several popular summary statistics both discrete and continuous under a variety of common settings we assume a flexible linear growth curve model to describe the repeated measures responses and focus on the types of settings that commonly arise in hiv aids and other diseases

line graphs link partitions and overlapping communities in this paper we use a partition of the links of a network in order to uncover its community structure this approach allows for communities to overlap at nodes so that nodes may be in more than one community we do this by making a node partition of the line graph of the original network in this way we show that any algorithm that produces a partition of nodes can be used to produce a partition of links we discuss the role of the degree heterogeneity and propose a weighted version of the line graph in order to account for this

modelling the collapse transition in branched polymers abstract we consider a number of lattice animal models for the collapse transition exhibited by branched polymers in dilute solution in all models the collapse is driven by either some kind of near neighbour fugacity or cycle fugacity we describe rigorous upper and lower bounds on the temperature dependence of the limiting free energy and compare these bounds with numerical estimates derived using new exact enumeration data finally we comment on the difficulties associated with the interpretation of results for the specific heat

ferromagnetism in the hubbard model on line graphs and further considerations let l g be the line graph of a graph g v e the hubbard model on l g has ferromagnetic ground states with a saturated spin if the interaction is repulsive u 0 and if the number of electrons n satisfies n or m m mod e mod mod v mod 1 if g is bipartite and m mod e mod mod v mod otherwise the author shows that the ferromagnetic ground state is unique if n m further he gives a sufficient condition for the existence of other ground states if n m the results are valid also for a multi band hubbard model on a bipartite graph in the case of a periodic lattice the results are related to the existence of a flat energy band

on graph entropy measures based on the number of independent sets and matchings abstract in this paper we consider the graph entropy measures based on the number of independent sets and matchings the reason to study these measures relates to the fact that the independent set and matching problem is computationally demanding however these features can be calculated for smaller networks in case one can establish efficient estimations those measures may be also used for larger graphs so we establish some upper and lower bounds as well as some information inequalities for these information theoretic quantities in order to give further evidence we also generate numerical results to study these measures such as list the extremal graphs for these entropies those results reveal the two entropies possess some new features

1 d expansions for critical amplitudes the author shows how to expand critical amplitudes in inverse powers of the dimensionality d the technique which is rather general is applied to the classical n vector model with particular emphasis on the self avoiding walk n 0 and ising n 1 cases the existence of these expansions and of similar ones for critical points and critical constants suggests that 1 d expansions for non universal quantities are analogous to the epsilon expansions for universal quantities

a note on walk entropies in graphs abstract the concept of walk entropy of a graph has been recently introduced in e estrada et al 2014 4 in that paper the authors formulated two conjectures about walk entropies in the present note we prove the first of these two conjectures and propose a stronger form of it

1 d expansions for the free energy of weakly embedded site animal models of branched polymers we give the 1 d expansion to order 1 d5 for the limiting reduced free energy of a weakly embedded two variable site animal model of branched polymers hence we are able to derive expansions for the free energy of related one variable models and the growth constant of weakly and strongly embedded trees again to order 1 d5 we argue that the free energy expansions are asymptotically correct for a small range of fugacities only exact results show that animals with a compact hypercubic structure make the overwhelming contribution to the total number of weakly embedded site animals especially for large d this result is not reflected in the free energy 1 d expansions where trees make the dominant contribution we also derive new exact enumeration data for lattice animals using intermediate calculations in the derivation of the 1 d expansions thus we give partition functions for the general d dimensional hypercubic lattice for one to 13 sites

walk entropy and walk regularity abstract a graph is said to be walk regular if for each l 1 every vertex is contained in the same number of closed walks of length l we construct a 24 vertex graph h 4 that is not walk regular yet has maximized walk entropy s v h 4 log 24 for some 0 this graph is a counterexample to a conjecture of benzi 2014 1 conjecture 3 1 we also show that there exist infinitely many temperatures 0 0 so that s v g 0 log n g if and only if a graph g is walk regular

the problem of deficiency indices for discrete schrodinger operators on locally finite graphs the number of self adjoint extensions of a symmetric operator acting on a complex hilbert space is characterized by its deficiency indices given a locally finite unoriented simple tree we prove that the deficiency indices of any discrete schrodinger operator are either null or infinite we also prove that all deterministic discrete schrodinger operators which act on a random tree are almost surely self adjoint furthermore we provide several criteria of essential self adjointness we also address some importance to the case of the adjacency matrix and conjecture that given a locally finite unoriented simple graph its deficiency indices are either null or infinite besides that we consider some generalizations of trees and weighted graphs

communicability in complex networks we propose a new measure of the communicability of a complex network which is a broad generalization of the concept of the shortest path according to the new measure most of the real world networks display the largest communicability between the most connected popular nodes of the network assortative communicability there are also several networks with the disassortative communicability where the most popular nodes communicate very poorly to each other using this information we classify a diverse set of real world complex systems into a small number of universality classes based on their structure dynamic correlation in addition the new communicability measure is able to distinguish finer structures of networks such as communities into which a network is divided a community is unambiguously defined here as a set of nodes displaying larger communicability among them than to the rest of the nodes in the network

lattice animal contact models of a collapsing branched polymer the authors consider two lattice animal models for the collapse of dilute branched polymers in a good solvent in both cases the collapse is driven by a near neighbour contact fugacity the two models differing in the way molecular weights are assessed either by the site content or the bond content of the animal they describe some rigorous results including bounds for the temperature dependence of their reduced limiting free energies on a d dimensional hypercubic lattice and compare these results with numerical estimates derived from exact enumeration data from the specific heat they estimate the collapse temperature of both models on a variety of lattices in addition they estimate the cross over exponent phi and find that for both models phi 0 60 or 0 03 d 2 and phi 0 82 or 0 02 d 3

the cluster structure in collapsing animals in this paper i revisit the connection between edge percolation and the collapse transition in lattice animals it was shown by domb 1976 j phys a math gen 9 l141 that the critical percolation point is a transition in a certain model of self interacting animals i extend this result by showing that the free energy of lattice animals in the cycle contact model is non analytic in contact and cycle activities at points other than the critical percolation point this correspondence between percolation and collapsing animals suggests that the collapse transition in this model may be related to percolation contact collapse in animals is then studied from a percolation of clusters of contacts perspective i first argue that there is a critical activity for percolation of clusters of contacts and then investigate this numerically

the efficient determination of the percolation threshold by a frontier generating walk in a gradient the frontier in gradient percolation is generated directly by a type of self avoiding random walk the existence of the gradient permits one to generate an infinite walk on a computer of finite memory from this walk the percolation threshold pc for a two dimensional lattice can be determined with apparently maximum efficiency for a naive monte carlo calculation or n 12 for a square lattice the value pc 0 592745 or 0 000002 is found for a simulation of n 2 6 1011 total steps occupied and blocked perimeter sites the power of the method is verified on the kagome site percolation case

universality in branched polymers on d dimensional hypercubic lattices the authors have derived series for weakly and strongly embeddable trees in d dimensional simple hypercubic lattices for arbitrary integral d for d 2 3 9 they present series evidence that such trees are in the same universality class as lattice animals in addition they have derived expansions in inverse powers of sigma 2d 1 for the growth parameters for bond and site trees and compare these with the corresponding results for animals

the collapse transition for the lattice trees the authors consider a lattice tree model for the collapse of dilute branched polymers in the good solvent regime in which the collapse is driven by a near neighbour contact fugacity they describe some rigorous results including bounds for the temperature dependence of the reduced limiting free energy and compare these results with numerical estimates derived from exact enumeration data from the specific heat they estimate the collapse temperature tc and the cross over exponent phi 0 in two and three dimensions they find phi 0 0 60 or 0 03 d 2 and phi 0 0 82 or 0 03 d 3 finally they speculate on a possible roughening transition which may occur at a temperature tr tc

a study of uniform stars using 1 d expansions and numerical methods we study a lattice model of an interacting uniform self avoiding star polymer with f branches a 1 d expansion for the limiting reduced free energy is derived through order 1 d for general f and for f 3 to order 1 d 2 the terms in the expansion are independent of f and agree term by term with the corresponding expansion for interacting self avoiding walks we also present a miscellany of numerical results obtained by more conventional series and monte carlo techniques all our results both past and present support the conjecture that the limiting reduced free energies of f stars walks and polygons are identical for all values of the interaction parameter

a history of graph entropy measures this survey seeks to describe methods for measuring the entropy of graphs and to demonstrate the wide applicability of entropy measures setting the scene with a review of classical measures for determining the structural information content of graphs we discuss graph entropy measures which play an important role in a variety of problem areas including biology chemistry and sociology in addition we examine relationships between selected entropy measures illustrating differences quantitatively with concrete examples

stratification of summary statistic tests according to missing data patterns summary statistics such as slope or area under the time response curve reduce the dimensionality of repeated measures data and can thereby simplify the comparison of groups in longitudinal studies since summary statistic distributions vary according to the amount timing and type of any missingness that occurs one must choose between analysing the data unconditionally or conditionally on the missingness patterns this paper uses simulations to compare such unstratified and stratified summary statistic analyses with respect to their size and power under models that allow for both non informative and informative missingness mechanisms of particular interest is the robustness of these methods to violations of the assumptions that one must make if they are to have proper test size it is found that stratification of the analysis tends to result in an increase of power and improves the robustness to violations of missing data assumptions

entropy of network ensembles in this paper we generalize the concept of random networks to describe network ensembles with nontrivial features by a statistical mechanics approach this framework is able to describe undirected and directed network ensembles as well as weighted network ensembles these networks might have nontrivial community structure or in the case of networks embedded in a given space they might have a link probability with a nontrivial dependence on the distance between the nodes these ensembles are characterized by their entropy which evaluates the cardinality of networks in the ensemble in particular in this paper we define and evaluate the structural entropy i e the entropy of the ensembles of undirected uncorrelated simple networks with given degree sequence we stress the apparent paradox that scale free degree distributions are characterized by having small structural entropy while they are so widely encountered in natural social and technological complex systems we propose a solution to the paradox by proving that scale free degree distributions are the most likely degree distribution with the corresponding value of the structural entropy finally the general framework we present in this paper is able to describe microcanonical ensembles of networks as well as canonical or hidden variable network ensembles with significant implications for the formulation of network constructing algorithms

maximum walk entropy implies walk regularity the notion of walk entropy sv g sv g for a graph g at the inverse temperature was put forward recently by estrada et al 2014 7 it was further proved by benzi 1 that a graph is walk regular if and only if its walk entropy is maximum for all temperatures i i where i is a set of real numbers containing at least an accumulation point benzi 1 conjectured that walk regularity can be characterized by the walk entropy if and only if there is a 0 0 such that sv g sv g is maximum here we prove that a graph is walk regular if and only if the view the mathml sourcesv g 1 lnn we also prove that if the graph is regular but not walk regular view the mathml sourcesv g 0 0 and view the mathml sourcelim 0sv g lnn lim sv g if the graph is not regular then view the mathml sourcesv g lnn for every 0 0 for some 0 0

families of line graphs and their quantization any directed graph g with n vertices and j edges has an associated line graph l g where the j edges form the vertices of l g we show that the non zero eigenvalues of the adjacency matrices are the same for all graphs of such a family ln g we give necessary and sufficient conditions for a line graph to be quantisable and demonstrate that the spectra of associated quantum propagators follow the predictions of random matrices under very general conditions line graphs may therefore serve as models to study the semiclassical limit of large matrix size of a quantum dynamics on graphs with fixed classical behaviour

borel type bounds for the self avoiding walk connective constant let be the self avoiding walk connective constant on we show that the asymptotic expansion for c 1 in powers of 1 2d satisfies borel type bounds this supports the conjecture that the expansion is borel summable

lattice trees with specified topologies the authors study the total numbers of lattice trees with specified topologies for strongly embeddable or site clusters with ni branching points of degree i they show how to prove rigorously that the growth constants exist and are all equal to the neighbour avoiding walk limit nu they derive some exact upper bounds for the critical exponents associated with the star comb and brush topologies exact enumeration data are derived and analysed for both weak and strong embeddings of some stars combs and brushes on the square triangular simple cubic and d 4 simple hypercubic lattices using the exact enumeration data for the general d dimensional simple hypercubic lattice and the exact results for the interior of a bethe lattice they derive expansions for the growth constants in inverse powers of the dimensionality these results are consistent with the growth constants being equal to the appropriate walk limits mu or nu

asymptotic behavior of linear permutation tests under general alternatives with application to test selection and study design abstract tests based on the permutation of observations are a common and attractive method of comparing two groups of outcomes in part because they retain proper test size with minimal assumptions and can have high efficiency toward specific alternatives of interest in addition permutation tests may be used with discrete or categorical outcomes for which linear rank tests are not designed permutation tests are now increasingly used to analyze discrete or continuous responses that themselves are functions of several statistics examples of such summary statistics include the area under the curve generated by repeated measures of a laboratory marker or an overall composite score from a quality of life study here even simple structures for the joint distribution of the component statistics can lead to complex differences between the distributions of summary statistics of the comparison groups despite their attractive features surprisingly little is known about the behavior of linear permutation tests

walk entropies in graphs abstract entropies based on walks in graphs and in their line graphs are defined they are based on the summation over diagonal and off diagonal elements of the exponential of the adjacency matrix known as the network communicability the walk entropies are strongly related to the walk regularity of graphs and line graphs they are not biased by the graph size and have significantly better correlation with the inverse participation ratio of the eigenmodes of the adjacency matrix than other graph entropies a homogeneous weighting of the edges of the graph is used to simulate the effects of the temperature over the entropies defined in particular the walk entropy of graphs is non monotonic for regular but non walk regular graphs in contrast to non regular graphs

central limit theorems for large graphs method of quantum decomposition a new method is proposed for investigating spectral distribution of the combinatorial laplacian adjacency matrix of a large regular graph on the basis of quantum decomposition and quantum central limit theorem general results are proved for cayley graphs of discrete groups and for distance regular graphs the coxeter groups and the johnson graphs are discussed in detail by way of illustration in particular the limit distributions obtained from the johnson graphs are characterized by the meixner polynomials which form a one parameter deformation of the laguerre polynomials

marriage of exact enumeration and 1 d expansion methods lattice model of dilute polymers we consider the properties of a self avoiding polymer chain with nearestneighbor contact energy on ad dimensional hypercubic lattice general theoretical arguments enable us to prescribe the exact analytic form of then segment chain partition functionc n and unknown coefficients for chains of up to 11 segments are determined using exact enumeration data ind 2 6 this exact form provides the main ingredient to produce a large n expansion ind 1of the chain free energy through fifth order with the full dependence on the contact energy retained the dependent chain connectivity constant and free energy amplitude are evaluated within thed 1expansion too d 5 our general formulation includes for the first time self avoiding walks neighboravoiding walks theta and collapsed chains as particular limiting cases

graph entropy based on the number of spanning forests of c cyclic graphs abstract graph entropies have been introduced to quantitatively measure the structural information content of graphs and networks they have plenty of applications in various fields utilizing the number of subgraphs to establish measures for determining the complexity of molecular graphs are also prevalent in the study of mathematical chemistry in this paper we develop a new graph entropy measure that is based on the number of spanning forests we prove explicit expressions for the entropy for trees unicyclic and bicyclic graphs and show that the cycle graph cn attains the maximal value of the entropy for unicyclic graphs with order n and large cycle lengths based on generating numerical results we conjecture extremal unicyclic graphs with respect to the entropy as well as we compare the values of our entropy for c cyclic graphs and generate graphs of bicyclic graphs and tricyclic graphs with 6 vertices for performing further research