bounds for the matching number and cyclomatic number of a signed graph in terms of rank abstract a signed graph g is a graph with a sign attached to each of its edges where g is the underlying graph of g let m g c g and r g be the matching number the cyclomatic number and the rank of the adjacency matrix of g respectively in this paper we investigate the relation among the rank the matching number and the cyclomatic number of a signed graph and prove that 2 m g 2 c g r g 2 m g c g furthermore signed graphs reaching the lower bound or the upper bound are respectively characterized

factorization of partitions in two dimensional lattices and complexity estimates connected partitions in two dimensional lattices naturally arise in studying the cluster configurations of a large class of dynamical systems we introduce a factorization into dichotomic factors allowing a meaningful implementation of the partition algebra this factorization proves useful in particular in the reduction process between couples of partitions the time series approach to the complexity based on the metric properties of the partition spaces may consequently be extended from one to two dimensional processes we check the efficiency of this formalism on the spin system simulated by the q2r automaton

on the sum of laplacian eigenvalues of a signed graph abstract for a signed graph let e denote the number of edges and s k denote the sum of the k largest eigenvalues of the laplacian matrix of we conjecture that for any signed graph with n vertices s k e k 1 2 1 holds for k 1 n we prove the conjecture for any signed graph when k 2 and prove that this conjecture is true for unicyclic and bicyclic signed graphs

mixed paths and cycles determined by their spectrum abstract a mixed graph is obtained from a graph by orienting some of its edges the hermitian adjacency matrix of a mixed graph with the vertex set v 1 v n is the matrix h h i j n n where h i j h j i i if there is a directed edge from v i to v j h i j 1 if there exists an undirected edge between v i and v j and h i j 0 otherwise the hermitian spectrum of a mixed graph is defined to be the spectrum of its hermitian adjacency matrix in this paper we study mixed graphs which are determined by their hermitian spectrum dhs first we show that each mixed cycle is switching equivalent to either a mixed cycle with no directed edges c n a mixed cycle with exactly one directed edge c n 1 or a mixed cycle with exactly two consecutive directed edges with the same direction c n 2 and we determine the spectrum of these three types of cycles next we characterize all dhs mixed paths and mixed cycles we show that all mixed paths of even order except p 8 and p 14 are dhs it is also shown that mixed paths of odd order except p 3 are not dhs also all cospectral mates of p 8 p 14 and p 4 k 1 and two families of cospectral mates of p 4 k 3 where k 1 are introduced finally we show that the mixed cycles c 2 k and c 2 k 2 where k 3 are not dhs but the mixed cycles c 4 c 4 2 c 2 k 1 c 2 k 1 2 c 2 k 1 1 and c 2 j 1 except c 7 1 c 9 1 c 12 1 and c 15 1 are dhs where k 1 and j 2

metric features of self organized criticality states in sandpile models a new characterization of self organized criticality soc states is developed by using metric features of the configuration s space quantities mainly referring to the partition formalism as mutual factorization shannon entropy and rohlin distances with their distributions and power spectra are considered time series for these observables give account of geometrical and dynamical complexity through the interdependence of fractality and flicker noise for bak tang wiesenfeld and manna automata new indicators enforce previous results given by standard parameters and allow a deeper insight into the structure of soc configurations and their time behaviour moreover we obtain indications regarding a possible split in the universality class of the two automata

signed bicyclic graphs minimizing the least laplacian eigenvalue abstract a signed graph is a pair g where g v g e g is a graph and e g 1 1 is the sign function on the edges of g for a signed graph we consider the laplacian matrix defined as l d g a where d g is the matrix of vertices degrees of g and a is the signed adjacency matrix the least laplacian eigenvalue is zero if and only if the signed graph is balanced i e all cycles contain an even number of negative edges here we show that among the unbalanced bicyclic signed graphs of given order n 5 the least laplacian eigenvalue is minimal for signed graphs consisting of two triangles only one of which is unbalanced connected by a path we also identify the signed graphs minimizing the least eigenvalue among those whose unbalanced bicyclic base is a theta graph

universality in sandpile models a classification of sandpile models into universality classes is presented on the basis of extensive numerical simulations in which we measure an extended set of exponents the manna two state model s s manna j phys a 24 l363 1991 is found to belong to a universality class of random neighbor models which is distinct from the universality class of the original model of bak tang and wiesenfeld p bak c tang and k wiesenfeld phys rev lett 59 381 1987 directed models are found to belong to a universality class which includes the directed model introduced and solved by dhar and ramaswamy d dhar and r ramaswamy phys rev lett 63 1659 1989

disturbance decoupling for linear time invariant systems a matrix pencil approach we give a new systematic analysis of disturbance decoupling problems for standard linear time invariant systems based on the theory of matrix pencils this approach is based on the computation of condensed forms under orthogonal equivalence transformations from these forms which can be computed in a numerically stable way we obtain new necessary and sufficient conditions that are numerically verifiable and furthermore we immediately obtain numerically stable algorithms to compute the desired compensators we present a numerical example that demonstrates the properties of the new approach

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

inclusion principle for linear time varying systems the main objective of this paper is to consider time varying linear systems within the framework of the inclusion principle starting from general definitions conditions are derived for a dynamic system to include a dynamic system of smaller dimension particular attention is paid to restriction and aggregation the most important special cases of inclusion it is proved using geometric arguments within the general time varying context that any inclusion relationship on a given time interval can be decomposed into a sequence of restriction aggregation pairs connections between inclusion and zero state equivalence are discussed the paper also presents a formulation of the contractibility expandability conditions for time varying state feedback controllers

integral signed subcubic graphs abstract a signed graph is called integral if all eigenvalues of its adjacency matrix are integers it is well known that there are thirteen connected integral cubic graphs in this paper we determine all connected integral signed subcubic graphs among these signed graphs there are two classes of integral signed cubic graphs one contains an infinite family of connected integral signed cubic graphs and another contains 14 sporadic connected integral signed cubic graphs

feedback design for regularizing descriptor systems this paper surveys numerical techniques for the regularization of descriptor generalized state space systems by proportional and derivative feedback we review generalizations of controllability and observability to descriptor systems along with definitions of regularity and index in terms of the weierstras canonical form three condensed forms display the controllability and observability properties of a descriptor system the condensed forms are obtained through orthogonal equivalence transformations and rank decisions so they may be computed by numerically stable algorithms in addition the condensed forms display whether a descriptor system is regularizable i e when the system pencil can be made to be regular by derivative and or proportional output feedback and if so what index can be achieved also included is a a new characterization of descriptor systems that can be made to be regular with index 1 by proportional and derivative output feedback

generalized selection of complementary matrices in the inclusion principle this paper presents a strategy for choosing complementary matrices in the framework of the inclusion principle with state lq optimal control of lti systems it is based on translating the basic restrictions given by the inclusion principle into explicit block structures for these matrices the degree of freedom given by these structures is illustrated by means of an example of overlapping decentralized control design

contractibility of dynamic lti controllers using complementary matrices a generalized structure of complementary matrices involved in the input state output inclusion principle for linear time invariant systems including contractibility conditions for static state feedback controllers is well known in this note it is shown how to further extend this structure when considering contractibility of dynamic controllers necessary and sufficient conditions for contractibility are proved in terms of both unstructured and block structured complementary matrices for general expansion contraction transformation matrices explicit sufficient conditions for blocks of complementary matrices ensuring contractibility are proved for general expansion contraction transformation matrices moreover these conditions are further specialized for a particular class of transformation matrices

intermittency from collatz s itineraries and complexity indicators we introduce a new type of discrete map on positive reals showing peculiar kinds of complexity they combine a large scale component collatz s itineraries on integers with a small scale component standard chaotic systems such as logistic maps or random noise usual characterizations for example by lyapunov exponents prove senseless or deceptive other indicators are used and in particular a new entropy is introduced to avoid difficulties regarding invariant measures all computations converge on the conclusion that the crucial point in qualifying the type of complexity and predictability does not consist in the alternative between deterministic perturbation or random noise but in the peculiar way the large scale component reacts on the small scale component altering its genericity possible connections with systems of physical interest are indicated

decentralized estimation and control with overlapping input state and output decomposition abstract the extension principle which was introduced recently is generalized to the case of input state and output expansion estimator and controller design problems are considered within the framework of the extension principle decentralized estimator and controller design with overlapping decompositions is also discussed within the same framework it is shown that if the extension principle is used then any estimator or controller designed in the expanded spaces is contractible to the original spaces for implementation furthermore it is shown that if an estimator designed for the expanded system achieves good estimation and or asymptotic estimation then the contracted estimator also achieves good estimation and or asymptotic estimation for the original system similarly it is also shown that if a controller designed for the expanded system achieves stability and or good performance then the contracted controller achieves stability and or good performance for the original system

integer lu factorizations abstract let a be an m by n integer matrix and r rank a a necessary and sufficient condition is given for a to have an integer lu factorization and a modification of gaussian elimination is given for finding such factorizations when the first r leading principal minors are nonzero an algorithm is given for ordering the vertices of a tree with a loop at its root so that its adjacency matrix has an integer lu factorization

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

skew rank of an oriented graph in terms of matching number abstract an oriented graph g is a digraph without loops and multiple arcs where g is called the underlying graph of g let s g denote the skew adjacency matrix of g the rank of s g is called the skew rank of g denoted by sr g which is even since s g is skew symmetric li and yu 2015 12 proved that the skew rank of an oriented unicyclic graph g is either 2 m g 2 or 2 m g where m g denotes the matching number of g in this paper we extend this result to general cases it is proved that the skew rank of an oriented connected graph g is an even integer satisfying 2 m g 2 g sr g 2 m g where g e g v g 1 is the number of fundamental cycles also called the first betti number besides the oriented graphs satisfying sr g 2 m g 2 g are characterized definitely

relations between the inertia indices of a mixed graph and those of its underlying graph abstract given a graph g the mixed graph d g is obtained from g by orienting some of its edges where g is called the underlying graph of d g let p d g n d g resp p g n g be the positive inertia index and negative inertia index of d g resp g in this paper we first establish the inequalities d g p d g p g d g and d g n d g n g d g where d g is the dimension of cycle space of g furthermore all the corresponding extremal graphs are characterized

an inclusion principle for dynamic systems the purpose of this paper is to present a detailed study of the inclusion concept in dynamic systems which is a suitable mathematical framework for comparing systems with different dimensions the framework offers immediate results in reduced order modeling and the overlapping decentralized control of complex systems the presentation which is limited to linear constant systems relies on both the matrix algebra computations and the geometric elements structure to provide a balanced view of the issues involved in the concept of inclusion the framework is quite broad and has been used to consider nonlinear and time varying systems as well as systems with hereditary and stochastic effects

inclusion principle for descriptor systems the purpose of this paper is to propose an expansion contraction framework for linear constant descriptor systems within the inclusion principle for dynamic systems our primary objective is to provide an explicit characterization of the expansion process whereby a given descriptor system is expanded into the larger space where all its solutions are reproducible by the expanded descriptor system if appropriate initial conditions are selected when a control law is formulated in the expanded space the proposed characterizations provide contractibility conditions for implementation of the control law in the original system a full freedom is provided for selecting appropriate matrices in the proposed expansion contraction control scheme in particular the derived theoretical framework serves as a flexible environment for expansion contraction control design of descriptor systems under overlapping information structure constraints

minimum norm regularization of descriptor systems by mixed output feedback we study the regularization problem for linear constant coefficient descriptor systems ex ax bu y1 cx y2 x by proportional and derivative mixed output feedback necessary and sufficient conditions are given which guarantee that there exist output feedbacks such that the closed loop system is regular has index at most one and e bg has a desired rank i e there is a desired number of differential and algebraic equations to resolve the freedom in the choice of the feedback matrices we then discuss how to obtain the desired regularizing feedback of minimum norm and show that this approach leads to useful results in the sense of robustness only if the rank of e is decreased numerical procedures are derived to construct the desired feedback gains these numerical procedures are based on orthogonal matrix transformations which can be implemented in a numerically stable way

on the signless laplacian spectral characterization of the line graphs of lollipop graphs abstract let h g k n denote the lollipop graph on n vertices obtained by identifying a vertex of the cycle c g of order g and an end vertex of the path p k 1 of order k 1 in this paper we prove that all line graphs l h g k n of lollipop graphs are determined by their laplacian spectrum furthermore we show that all l h g k n but the case that g k 3 are determined by their signless laplacian spectrum and also give the q cospectral class of l h r r n r 3

spectral characterizations of signed cycles abstract a signed graph is a pair like g where g is the underlying graph and e g 1 1 is a sign function on the edges of g in this paper we study the spectral determination problem for signed n cycles c n with respect to the adjacency spectrum and the laplacian spectrum in particular for the laplacian spectrum we prove that balanced odd cycles and unbalanced cycles denoted respectively by c 2 n 1 and c n are uniquely determined by their laplacian spectra i e they are dls on the other hand we determine all laplacian cospectral mates of the balanced even cycles c 2 n so that we show that c 2 n is not dls the same problem is then considered for the adjacency spectrum hence we prove that odd signed cycles namely c 2 n 1 and c 2 n 1 are uniquely determined by their adjacency spectrum i e they are ds moreover we find cospectral mates for the even signed cycles c 2 n and c 2 n and we show that except the signed cycle c 4 even signed cycles are not ds and we provide almost all cospectral mates

contractibility of overlapping decentralized control abstract the purpose of this paper is to generalize the concept of contractibility of decentralized control laws in the inclusion principle after a system with overlapping subsystems is expanded into a larger space and decentralized control laws are formulated for the disjoint subsystems the laws need to be contracted for implementation in the original space we propose broader definitions of restriction and aggregation in the framework of inclusion which provide more flexibility in the contraction phase of the expansion contraction process in particular we discuss contractibility conditions for dynamic output controllers including state observers which have been of special interest in applications

on the characteristic polynomials and h ranks of the weighted mixed graphs abstract let d g be an n vertex weighted mixed graph with hermitian adjacency matrix h d g the characteristic polynomial of the weighted mixed graph d g is defined as d g det i n h d g r 0 n r n r and the h rank of d g written as r k d g is the rank of h d g in this paper we begin by interpreting all the coefficients of the characteristic polynomial d g then we establish recurrences for the characteristic polynomial d g by these obtained results as a unified approach some main results obtained in electron j combin 18 1 2011 156 linear algebra appl 436 2012 3597 3607 linear algebra appl 466 2015 182 207 can be deduced consequently furthermore for a weighted mixed bipartite cactus graph d g we give a necessary and sufficient condition for r k d g 2 t where t is any integer no greater than the matching number of g finally as an application of our results we study the minimum h rank problem determining the minimum h rank of bipartite cactus graphs

laplacian spectral characterization of roses a rose graph is a graph consisting of cycles that all meet in one vertex we show that except for two specific examples these rose graphs are determined by the laplacian spectrum thus proving a conjecture posed by liu and huang 2013 8 we also show that if two rose graphs have a so called universal laplacian matrix with the same spectrum then they must be isomorphic in memory of horst sachs 1927 2016 we show the specific case of the latter result for the adjacency matrix by using sachs theorem and a new result on the number of matchings in the disjoint union of paths

model abstraction and inclusion principle a comparison this paper discusses connections between the inclusion principle and model abstraction in large scale hierarchical control systems it also provides conditions for implementation of controllability and observability in linear continuous time dynamic systems satisfying either aggregation or restriction conditions as well as for observability implementation in linear continuous time dynamic systems satisfying conditions for abstraction

the characteristic polynomial and the matchings polynomial of a weighted oriented graph abstract let g be a weighted oriented graph with skew adjacency matrix s g then g is usually referred as the weighted oriented graph associated to s g denote by g the characteristic polynomial of the weighted oriented graph g which is defined as g det i n s g i 0 n a i g n i in this paper we begin by interpreting all the coefficients of the characteristic polynomial of an arbitrary real skew symmetric matrix in terms of its associated oriented weighted graph then we establish recurrences for the characteristic polynomial and deduce a formula on the matchings polynomial of an arbitrary weighted graph in addition some miscellaneous results concerning the number of perfect matchings and the determinant of the skew adjacency matrix of an unweighted oriented graph are given

metric features of a dipolar model the lattice spin model with nearest neighbour ferromagnetic exchange and dipolar interaction is studied by the method of time series for observables based on cluster configurations and associated partitions such as shannon entropy or hamming and rohlin distances previous results based on the two peaks shape of the specific heat suggested the existence of two possible transitions by the analysis of the shannon entropy and related indicators we obtain strong indications that the first one is a true phase transition corresponding to a particular melting process of oriented domains where coloured noise is present almost independently of true fractality the second one in contrast seems not to be a true transition it may be ascribed to a smooth balancing between two geometrical effects a progressive fragmentation of the big clusters possibly creating fractals and the slow onset of a small cluster chaotic phase comparison with the nearest neighbour ising ferromagnetic system reveals a substantial difference in the cluster geometrical properties of the two models and in their critical behaviour

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

the generalized eigenstructure problem in linear system theory the algebraic theory of linear time invariant systems has been studied in large detail during the past few decades and numerous computational algorithms have been developed to solve problems arising in this context in this paper the numerical aspects of a certain class of such algorithms dealing with what the author calls generalized eigenstructure problems are discussed some new and or modified algorithms are presented both the nmnerical stability of the algorithms and the conditioning of the problems they solve are analyzed using numerical criteria

signed graphs cospectral with the path abstract a signed graph is said to be determined by its spectrum if every signed graph with the same spectrum as is switching isomorphic with here it is proved that the path p n interpreted as a signed graph is determined by its spectrum if and only if n 0 1 or 2 mod 4 unless n 8 13 14 17 29 or n 3

signed graphs with cut points whose positive inertia indexes are two abstract a signed graph g consists of an underlying graph g and a sign function which assigns each edge uv of g a sign u v either positive or negative the adjacency matrix of g is defined as a g a u v with a u v u v a u v where a u v 1 if u v v g are adjacent and a u v 0 otherwise the positive inertia index of g written as p g is defined to be the number of positive eigenvalues of a g recently yu et al 2016 12 characterized the signed graphs g with pendant vertices such that p g 2 in this paper we extend the above work to a more general case characterizing the signed graphs g with cut points whose positive inertia index is 2

contractible controller design and optimal control with state and input inclusion abstract a new inclusion concept which involves both state and input expansion is introduced it is shown that with the approach undertaken here any control law designed in the expanded spaces is always contractible to the original spaces the inclusion of the cost functions is also discussed and necessary and sufficient conditions are derived it is shown that unlike the case when the two input spaces are identical the inclusion conditions must in general depend on the control laws however certain sufficient conditions which do not involve final control laws can be found if the inclusion is taken with respect to the optimal control finally controller design with overlapping decompositions is discussed within the framework of the new inclusion concept and optimal control

sandpile models with and without an underlying spatial structure we present a simple mean field model for the sandpile model introduced by bak tang and wiesenfeld btw phys rev lett 59 381 1987 in the mean field model we are able to pinpoint the process of self organization as well as the emerging scale invariance displayed as a power law distribution of avalanche sizes we discuss the btw sandpile model on a lattice and show that the dynamical behavior can be expressed as a transport problem this implies that the average avalanche size scales with the system size and additional heuristic arguments related to the transport properties more than indicate the origin of the power law behavior we review recent work in which scaling relations and additional constraints between the various critical exponents are addressed we demonstrate that some of the proposed relations are inconsistent we present a coherent theory in which the scaling relations along with additional constraints leave only one exponent unknown

a numerical solution for the simultaneous disturbance decoupling and row by row decoupling problem the numerical computation of the simultaneous disturbance decoupling and row by row decoupling problem is considered based on a condensed form for the system matrices the solvability of the simultaneous disturbance decoupling and row by row decoupling problem is reduced to the verification of an integer equality and the nonsingularity of a lower dimensional constant matrix the condensed form we use is computed using only orthogonal transformations which can be implemented in a numerically stable way hence it leads directly to an effective numerical method for solving the disturbance decoupling and row by row decoupling problem using existing tools such as lapack and matlab our result complements existing geometric and structural approaches which are of a theoretical nature and are not suitable for numerical computation

hermitian adjacency matrices and hermitian energies of mixed graphs abstract a complex adjacency matrix of a mixed graph is introduced in the present paper which is a hermitian matrix and called the hermitian adjacency matrix it incorporates both adjacency matrix of an undirected graph and skew adjacency matrix of an oriented graph some of its properties are studied furthermore properties of its characteristic polynomial are studied cospectral problems among mixed graphs including mixed graphs and their underlying graphs oriented graphs and their underlying graphs are studied we give equivalent conditions for a mixed graph especially oriented graph that share the same spectrum with its underlying graph as a consequence we reconfirm a conjecture which was proposed by cui and hou in ref 8 we also show that the spectrum of the hermitian matrix of a mixed graph is invariant when changing the value of any its cut edge if any correspondingly an energy of a mixed graph is introduced and called the hermitian energy it incorporates both the energy of an undirected graph and the skew energy of an oriented graph some of its bounds are given especially the mixed graphs with optimal upper bound of hermitian energy are characterized an infinite family of mixed graphs attaining the maximum hermitian energy is constructed moreover the hermitian energy of a mixed tree is showed to be equal to the energy of its underlying tree finally the integral formula for hermitian energy of a mixed graph is given

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

a note on the spectral characterization of dumbbell graphs the dumbbell graph denoted by da b c is a bicyclic graph consisting of two vertex disjoint cycles ca and cb joined by a path pc 3 c 1 having only its end vertices in common with the two cycles by using a new cospectral invariant for r r 1 almost regular graphs we will show that almost all dumbbell graphs without cycle c4 as a subgraph are determined by the adjacency spectrum

combinatorial approach for computing the characteristic polynomial of a matrix the study of the spectrum of weighted di graphs has attracted recently much attention in the literature here we use the coates digraph as a main tool to extend in a combinatorial way some well known results from the spectral graph theory on computing the characteristic polynomial of graphs new results are related to weighted di graph and thus to any square matrix

a canonical form for the inclusion principle of dynamic systems the inclusion principle provides a mathematical framework for comparing behavior of dynamic systems having different dimensions our main objective is to derive a canonical form for larger systems expansions that are obtained by expanding smaller systems contractions the form offers full freedom in selecting appropriate matrices for the expansion contraction process we will broaden the form to include feedback and propose an explicit characterization of contractible control laws subject to overlapping information structure constraints

spectral characterizations of lollipop graphs abstract the lollipop graph denoted by h n p is obtained by appending a cycle c p to a pendant vertex of a path p n p we will show that no two non isomorphic lollipop graphs are cospectral with respect to the adjacency matrix it is proved that for p odd the lollipop graphs h n p and some related graphs h n p are determined by the adjacency spectrum and that all lollipop graphs are determined by its laplacian spectrum

mixed graphs with h rank 3 abstract in the article mohar 2016 9 mohar determined all mixed graphs with h rank 2 and used it to classify cospectral graphs with respect to their hermitian adjacency matrix constructing a class of graphs which can not be determined by their h spectrum in the present paper we investigate the h rank of mixed graphs further determining the h ranks of those mixed graphs with trees cycles and complete bipartite graphs as underlying graphs respectively moreover we characterize all mixed graphs with h rank 3 and show that all connected mixed graphs with h rank 3 can be determined by their h spectrum

preservation of controllability observability in expanded systems the result contributed by the article is that controllability observability of an original continuous time lti dynamic system can always be simultaneously preserved in expanded systems within the inclusion principle when using block structured complementary matrices this new structure offers more degrees of freedom for the selection of specific complementary matrices than well known used cases such as aggregations and restrictions which enable such preservation only in certain special cases a complete unrestricted transmission of these qualitative properties from the original controllable observable system to its expansion is a basic requirement on the expansion contraction process mainly when controllers observers are designed in expanded systems to be consequently contracted for implementation in initially given systems an original system composed of two overlapped subsystems is adopted as a general prototype ease a numerical example is supplied

decentralized control and overlapping decomposition of mechanical systems part 1 system decomposition this paper improves a new inclusion concept called txtension to mechanical systems described by matrix second order equations recent work on specialization of the inclusion principle has been generalized to the expansion and contraction of input state and output spaces but without the possibility of constructing completely general contractible controllers in this paper necessary and sufficient conditions for extensions and contractible feedback controllers are proved they enable the construction of general contractible controllers without any constraint on the structure of controllers for mechanical systems

the rank of a graph after vertex addition abstract let r g denote the rank of the adjacency matrix of a graph g when a vertex and its incident edges are deleted from g the rank of the resultant graph cannot exceed r g and can decrease by at most 2 the problem of determining the exact effect of adding a single vertex to a graph is more difficult since the number of edges that can be added with this vertex is variable the rank of the new graph cannot decrease and it can increase by at most 2 we obtain results examining several cases of vertex addition

universality in sandpiles we perform extensive numerical simulations of different versions of the sandpile model we find that previous claims about universality classes are unfounded since the method previously employed to analyze the data suffered from a systematic bias we identify the correct scaling behavior and provide evidences suggesting that sandpiles with stochastic and deterministic toppling rules belong to the same universality class

the rank of a complex unit gain graph in terms of the matching number abstract a complex unit gain graph or t gain graph is a triple g t or g for short consisting of a simple graph g as the underlying graph of g the set of unit complex numbers t z c z 1 and a gain function e t with the property that e i j e j i 1 in this paper we prove that 2 m g 2 c g r g 2 m g c g where r g m g and c g are the rank of the hermitian adjacency matrix h g the matching number and the cyclomatic number of g respectively furthermore the complex unit gain graphs g t with r g 2 m g 2 c g and r g 2 m g c g are characterized these results generalize the corresponding known results about undirected graphs mixed graphs and signed graphs moreover we show that 2 m g v 0 r g 2 m g s holds for any s v g such that g s is bipartite and any subset v 0 of v g such that g v 0 is acyclic

decentralized control using quasi block diagonal dominance of transfer function matrices the purpose of this paper is to present a generalization of the nyquist array method to blockwise decompositions which is based upon a new version of block diagonal dominance an additional flexibility of the proposed method is in partitioning of the system matrices into disjoint as well as overlapping submatrices which increases considerably the class of control systems which can be designed via block diagonal dominance within this framework the controllers for individual subsystems can be designed independently of each other so that their union represents a decentralized controller for the overall system