computation theory of cellular automata self organizing behaviour in cellular automata is discussed as a computational process formal language theory is used to extend dynamical systems theory descriptions of cellular automata the sets of configurations generated after a finite number of time steps of cellular automaton evolution are shown to form regular languages many examples are given the sizes of the minimal grammars for these languages provide measures of the complexities of the sets this complexity is usually found to be non decreasing with time the limit sets generated by some classes of cellular automata correspond to regular languages for other classes of cellular automata they appear to correspond to more complicated languages many properties of these sets are then formally non computable it is suggested that such undecidability is common in these and other dynamical systems

maximum number of periodic orbits in parallel dynamical systems abstract for parallel dynamical systems over undirected graphs with a boolean maxterm or minterm functions as global evolution operators it is known that every periodic orbit has period less than or equal to two in fact periodic orbits of different periods cannot coexist and a fixed point theorem based in the uniqueness of a fixed point is also known in this paper we complete the study of the periodic structure of such systems providing a 2 periodic orbit theorem and giving an upper bound for the number of fixed points and also for the number of 2 periodic orbits actually we provide examples where these bounds are attained demonstrating that they are the best possible ones

research problems from the aveiro workshop on graph spectra abstract this is a collection of open problems presented at the aveiro workshop on graph spectra held at the university of aveiro portugal from april 10 12 2006

parallel discrete dynamical systems on independent local functions in this paper we extend the manner of defining the evolution update of discrete dynamical systems on boolean functions without limiting the local functions to being dependent restrictions of a global one then we analyze the cases concerned with parallel dynamical systems with the or and nand and nor functions as independent local functions over undirected and also directed dependency graphs this extension of the update method widely generalizes the traditional one where only a global boolean function is considered for establishing the evolution operator of the system besides our analysis allows us to show a richer dynamics in these new kinds of parallel dynamical systems

elements of a theory of simulation ii sequential dynamical systems we study a class of discrete dynamical systems that is motivated by the generic structure of simulations the systems consist of the following data a a finite graph y with vertex set 1 n where each vertex has a binary state b functions f i f 2 n f 2 n and c an update ordering p the functions f i update the binary state of vertex i as a function of the state of vertex i and its y neighbors and leave the states of all other vertices fixed the update ordering is a permutation of the y vertices by composing the functions f i in the order given by p one obtains the sequential dynamical system sds f y p email protected i 1nf p i f 2 n f 2 n we derive a decomposition result characterize invertible sds and study fixed points in particular we analyse how many different sds that can be obtained by reordering a given multiset of update functions and give a criterion for when one can derive concentration results on this number finally some specific sds are investigated

on the maximal index of graphs with a prescribed number of edges abstract among the graphs with a prescribed number of edges those with maximal index are determined the result confirms a conjecture of brualdi and hoffman

on the spectral radius of 0 1 matrices we determine the maximum spectral radius for 0 1 matrices with k2 andk2 1 1 s respectively and for symmetric 0 1 matrices with zero trace and e k21 s graphs with e edges in all cases equality is characterized

periods in xor parallel dynamical systems over directed dependency graphs abstract we analyze the periodic structure of parallel dynamical systems over directed dependency graphs whose evolution operator is the boolean function xor we prove that such systems can present periodic orbits of any period moreover we demonstrate that any kinds of periods can coexist at the same time in view of these results we study the periodic structure of these dynamical systems over complete digraphs complete bipartite digraphs acyclic digraphs and out trees

predecessors and garden of eden configurations in parallel dynamical systems on maxterm and minterm boolean functions abstract in this work we solve the classical predecessor problems for parallel dynamical systems on maxterm and minterm boolean functions actually we solve analytically the predecessor existence problem by giving a characterization to have a predecessor for any given configuration as a consequence we also get a characterization of the garden of eden configurations of these systems moreover the structure of the predecessors found out allows us to give a solution to the unique predecessor problem the coexistence of predecessors problem and the number of predecessors problem

on periods and equilibria of computational sequential systems abstract in this paper we show that sequential systems with boolean maxterms and minterms as global evolution operators can present orbits of any period besides we prove that periodic orbits with different periods greater than or equal to 2 can coexist nevertheless when a sequential dynamical system has fixed points we demonstrate that periodic orbits of other periods cannot appear finally we provide conditions to obtain a fixed point theorem in this context this work provides a relevant advance in the knowledge of the dynamics of such systems which constitute one of the most effective mathematical tools to model computational processes and other phenomena from other sciences moreover the ideas developed here could help to obtain similar results for other related systems

enumerating periodic orbits in sequential dynamical systems over graphs abstract it is well known that periodic orbits with any period can appear in sequential dynamical systems over undirected graphs with a boolean maxterm or minterm function as global evolution operator indeed fixed points cannot coexist with periodic orbits of greater periods while periodic orbits with different periods greater than 1 can coexist additionally a fixed point theorem about the uniqueness of a fixed point is known in this paper we provide an m periodic orbit theorem for m 1 and we give an upper bound for the number of fixed points and periodic orbits of period greater than 1 so completing the study of the periodic structure of such systems we also demonstrate that these bounds are the best possible ones by providing examples where they are attained

maximal graphs and graphs with maximal spectral radius abstract a maximal graph is a connected graph with degree sequence not majorized by the degree sequence of any other graph relationships between three different sequences of integers that describe maximal graphs are given recursive and explicit forms of the characteristic polynomial of the adjacency matrix of a maximal graph are derived a correspondence between maximal graphs and connected graphs with stepwise adjacency matrices shows that among all connected graphs with n vertices and e edges the graph with maximal spectral radius is a maximal graph such maximal graphs are identified for certain values of n and e

minimizing control variation in discrete time optimal control problems for a real practical system a large fluctuation in the control signal is highly undesirable to address this undesirable situation we investigate a discrete time optimal control problem subject to terminal state and all time step constraints on the state and control where the cost function is the sum of terminal cost and the variation of the control signal the variation of the control signal is expressed in terms of absolute value functions and hence is non smooth by a novel smooth transformation and the constraint transcription technique this problem is approximated by a constrained discrete time optimal control with the new cost function involves only smooth functions a gradient based computational method is then derived which is supported by rigorous convergence analysis two examples are provided to demonstrate the effectiveness and advantages of the proposed method we present a gradient based method for discrete time optimal control problems the cost function is the sum of terminal cost and the variation of control signal a smooth transformation and the constraint transcription technique are used two examples are provided to demonstrate the feasibility of the method

spectral radius of bipartite graphs abstract let k p q be positive integers with k p q 1 we prove that the maximum spectral radius of a simple bipartite graph obtained from the complete bipartite graph k p q of bipartition orders p and q by deleting k edges is attained when the deleted edges are all incident on a common vertex which is located in the partite set of order q our method is based on new sharp upper bounds on the spectral radius of bipartite graphs in terms of their degree sequences

new rough approximations for n cycles and n paths in this paper we continue the study of a finite simple undirected graph through the rough set techniques introduced by pawlak in the early 80 s in this sense we focus on the boolean information system induced from a circle or a line graph with n vertices and for such graphs we provide a complete description of the rough membership function and some results on the positive regions and the attribute dependency function

parallel and sequential dynamics of two discrete models of signed integer partitions abstract in this paper we complete and generalize some previous results concerning the computing of the sequential and parallel convergent time for two discrete dynamical system of signed integer partitions we also refine the concept of parallel convergent time for a finite graded partially ordered set briefly poset x which is also a discrete dynamical model to this aim we define the concept of fundamental sequence of x and we compute this sequence in two particularly important cases in the first case when x is the finite lattice s n r of all the signed integer partitions a r a 1 b 1 b n r such that r a r a 1 0 b 1 b n r n r where n r 0 and the unique part that can be repeated is 0 in the second case when x is the sub lattice s n d r of all the signed integer partitions of s n r having exactly d non zero parts the relevance of the previous lattices as discrete dynamical models is related to their link with some unsolved extremal combinatorial sum problems

sequential dynamical systems over words this paper is motivated by the theory of sequential dynamical systems sds developed as a basis for a mathematical theory of computer simulation a sequential dynamical system is a collection of symmetric boolean local update functions with the update order determined by a permutation of the boolean variables in this paper the notion of sds is generalized to allow arbitrary functions over a general finite field with the update schedule given by an arbitrary word on the variables the paper contains generalizations of some of the known results about sds with permutation update schedules in particular an upper bound on the number of different sds over words of a given length is proved and open problems are discussed

a bound on the spectral radius of graphs abstract let g be a simple connected graph with n vertices and m edges let a be the adjacency matrix of g the spectral radius a of a satisfies a 2m n 1 with equality if and only if g is isomorphic to one of the following two graphs a the star k1 n 1 b the complete graph kn

graph irregularity and its measures abstract let g be a simple graph if all vertices of g have equal degrees then g is said to be regular otherwise g is irregular there were various attempts to quantify the irregularity of a graph of which the collatz sinogowitz index bell index albertson index and total irregularity are the best known we now show that no two of these irregularity measures are mutually consistent namely that for any two such measures irrx and irry there exist pairs of graphs g1 g2 such that irrx g1 irrx g2 but irry g1

the maximal eigenvalue of 0 1 matrices with prescribed number of ones abstract we determine the maximum spectral radius for 0 1 matrices with m 2 l ones for l 2 m 2 m 3 for all m and for a fixed l and m m l similar results are obtained for symmetric 0 1 matrix with zero diagonal in all cases equality is characterized

on the spectral radius of tricyclic graphs with a maximum matching abstract in this paper we determine the graph with the largest spectral radius among all the tricyclic graphs with a maximum matching of cardinality m m 8

a palindromization map for the free group we define a self map pal f 2 f 2 of the free group on two generators a b using automorphisms of f 2 that form a group isomorphic to the braid group b 3 the map pal restricts to de luca s right iterated palindromic closure on the submonoid generated by a b we show that pal is continuous for the profinite topology on f 2 it is the unique continuous extension of de luca s right iterated palindromic closure to f 2 the values of pal are palindromes and coincide with the elements g f 2 such that abg and bag are conjugate

on the maximal index of connected graphs abstract let h n e denote the set of all connected graphs having n vertices and e edges the graphs in h n n k with maximal index are determined for k of form r 2 1 and n arbitrary

a note on the irregularity of graphs abstract denote by 1 g the largest eigenvalue of a real 0 1 adjacency matrix of a graph g and by d g the mean degree of g collatz and sinogowitz proposed 1 g d g as a measure of irregularity of g a second such measure is the variance of the vertex degrees of g the most irregular graphs according to these measures are determined for certain classes of graphs and the two measures are shown to be incompatible for some pairs of graphs

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

on the index of tricyclic graphs with perfect matchings abstract let t 2 k be the set of all tricyclic graphs on 2 k k 2 vertices with perfect matchings in this paper we discuss some properties of the connected graphs with perfect matchings and then determine graphs with the largest index in t 2 k

palindromic primitives and palindromic bases in the free group of rank two abstract the present paper records more details of the relationship between primitive elements and palindromes in f 2 the free group of rank two we characterize the conjugacy classes of palindromic primitive elements as those in which cyclically reduced words have odd length we identify large palindromic subwords of certain primitives in conjugacy classes which contain cyclically reduced words of even length we show that under obvious conditions on exponent sums pairs of palindromic primitives form palindromic bases for f 2 further we note that each cyclically reduced primitive element is either a palindrome or the concatenation of two palindromes

variable neighborhood search for extremal graphs 16 some conjectures related to the largest eigenvalue of a graph abstract we consider four conjectures related to the largest eigenvalue of the adjacency matrix of a graph i e to the index of the graph three of them have been formulated after some experiments with the programming system autographix designed for finding extremal graphs with respect to given properties by the use of variable neighborhood search the conjectures are related to the maximal value of the irregularity and spectral spread in n vertex graphs to a nordhaus gaddum type upper bound for the index and to the maximal value of the index for graphs with given numbers of vertices and edges none of the conjectures has been resolved so far we present partial results and provide some indications that the conjectures are very hard

graphs with maximal signless laplacian spectral radius abstract by the signless laplacian of a simple graph g we mean the matrix q g d g a g where a g d g denote respectively the adjacency matrix and the diagonal matrix of vertex degrees of g it is known that connected graphs g that maximize the signless laplacian spectral radius q g over all connected graphs with given numbers of vertices and edges are degree maximal for a maximal graph g with n vertices and r distinct vertex degrees r r 1 1 it is proved that q g q h for some maximal graph h with n 1 respectively n vertices and the same number of edges as g if either g has precisely two dominating vertices or there exists an integer i 2 i r 2 respectively if there exist positive integers i l with l 2 i r 2 such that i r 1 i n 1 respectively i r 1 i l r l 1 graphs that maximize q g over the class of graphs with m edges and m k vertices for k 0 1 2 3 are completely determined

dichotomy results for fixed point counting in boolean dynamical systems we present dichotomy theorems regarding the computational complexity of counting fixed points in boolean discrete dynamical systems i e finite discrete dynamical systems over the domain 0 1 for a class f of boolean functions and a class g of graphs an f g system is a boolean dynamical system with local transitions functions lying in f and graphs in g we show that if local transition functions are given by lookup tables then the following complexity classification holds let f be a class of boolean functions closed under superposition and let g be a graph class closed under taking minors if f contains all min functions all max functions or all self dual and monotone functions and g contains all planar graphs then it is p complete to compute the number of fixed points in an f g system otherwise it is computable in polynomial time we also prove a dichotomy theorem for the case that local transition functions are given by formulas over logical bases this theorem has a significantly more complicated structure than the theorem for lookup tables a corresponding theorem for boolean circuits coincides with the theorem for formulas

palindromic width of free nilpotent groups abstract in this paper we consider the palindromic width of free nilpotent groups in particular we prove that the palindromic width of a finitely generated free nilpotent group is finite we also prove that the palindromic width of a free abelian by nilpotent group is finite

bounds on the spectral radius of graphs with e edges abstract we give upper bounds for the spectral radius of a graph with e edges provided that there is no complete graph with e edges our bounds are sharp for i the complete graphs with one two or three edges removed ii the complete graph with one added vertex and edge

elements of a theory of simulation iii equivalence of sds in the development of mathematical foundations for a theory of simulation a certain class of discrete sequential dynamical systems sds is of particular importance these systems which we refer to as sds consist of a a graph y with vertex set 1 2 n where each vertex has associated a binary state b a vertex labeled set of functions f i y f 2 n f 2 n and c a permutation p s n the function f i y update the state of vertex i as a function of the states of vertex i and its y neighbors and leaves all other states invariant by composing these functions in the order given by p we obtain the sds f y p i 1 nf p i y f 2 n f 2 n in this paper we give a combinatorial upper bound for the number of non equivalent sds for a given graph and we compute this bound explicitly for certain classes of graphs we give a full characterization of invertible sds and analyze the set of fixed points of sequential and parallel cellular automata further we introduce the concept of maj type sds and show that systems of this class only have fixed points as attractors finally we analyze sds that are fixed point free for arbitrary base graphs

elements of a theory of computer simulation i sequential ca over random graphs this paper is a first step in the development of mathematical foundations for a theory of simulation we employ sequentially updated cellular automata sca over arbitrary graphs as a paradigmatic framework in the development of the theory we focus on the properties of causal dependencies among local mappings in a simulation let y be a graph in which a each vertex i has a state x i0 1 and b there exists a local map i defined on the states of the y neighbors and x i that returns the state of y i0 1 suppose i 1 i n is a permutation of the y vertices then the order of application of the local maps i k induces a sequential or asynchronous cellular automaton sca over y in this paper we introduce a mapping between the base graph y over which the sca is defined and which represents the mutual dependencies of the local maps and the update graph u y whose vertices are permutations of all y vertices two permutations p 1 p 2 are adjacent in u y if they differ in exactly two consecutive coordinates i k i k 1 and i k i k 1 ey u y represents a conceptual framework which allows to determine equivalence classes of sca for a given graph x independent of the family of local maps i 1 i n we consider a u as a random variable over g n p and analyze the induced random graph u g n p the update graph and show b that u exhibits properties of a covariant functor

parallel dynamical systems over directed dependency graphs abstract in this work we analyze the orbit structure of parallel discrete dynamical systems over directed dependency graphs with boolean functions as global functions in this sense for the cases corresponding to the simplest boolean functions and and or it is proved that only fixed or eventually fixed points appear as occurs over undirected dependency graphs however for general boolean functions it is shown that any period can appear so breaking the pattern found for the undirected case where only eventually fixed points or 2 periodic orbits can exist