classification of rings with projective zero divisor graphs abstract let r be a commutative ring and r denote its zero divisor graph in this paper we investigate the crosscap number of the non orientable compact surface which r can be embedded and illustrate all finite commutative rings r up to isomorphism such that r is projective

complemented zero divisor graphs and boolean rings abstract for a commutative ring r the zero divisor graph of r is the graph whose vertices are the nonzero zero divisors of r such that the vertices x and y are adjacent if and only if x y 0 in this paper we classify the zero divisor graphs of boolean rings as well as those of boolean rings that are rationally complete we also provide a complete list of those rings whose zero divisor graphs have the property that every vertex is either an end or adjacent to an end

the zero divisor graph of a commutative ring for each commutative ring r we associate a simple graph r we investigate the interplay between the ring theoretic properties of r and the graph theoretic properties of r

planar zero divisor graphs abstract this paper answers the question of anderson frazier lauve and livingston for which finite commutative rings r is the zero divisor graph r planar we build upon and extend work of akbari maimani and yassemi who proved that if r is any local ring with more than 32 elements and r is not a field then r is not planar they left open the question is it true that for any local ring r of cardinality 32 which is not a field r is not planar in this paper we answer this question in the affirmative we prove that if r is any local ring with more than 27 elements and r is not a field then r is not planar moreover we determine all finite commutative local rings whose zero divisor graph is planar

the group of automorphisms of a zero divisor graph based on rank one upper triangular matrices abstract let f q be a finite field with q elements n 2 a positive integer mat n q the set of all n n matrices over f q r n q the set of all rank one upper triangular matrices in mat n q the zero divisor graph of mat n q written as mat n q is a directed graph with vertex set all nonzero zero divisors of mat n q and there is a directed edge from a vertex a to a vertex b written as a b if and only if a b 0 in this paper we determine the automorphisms of an induced subgraph written as r n q of mat n q with vertex set r n q the main theorem of this article proves that a bijective map on r n q with n 3 is an automorphism of r n q if and only if x a x p 1 x i j p x x i j r n q where a x f q depends on x p is an invertible upper triangular matrix is an automorphism of the field f q x i j denotes the matrix whose i j entry is x i j

boolean rings and reciprocal eigenvalue properties abstract let a be the adjacency matrix of the zero divisor graph r of a finite commutative ring r containing nonzero zero divisors in this paper it is shown that r is the zero divisor graph of a boolean ring if and only if det a 1 also a is similar to plus or minus its inverse whenever r is a boolean ring as a consequence it is proved that r is the zero divisor graph of a boolean ring if and only if the set of eigenvalues including multiplicities of r can be partitioned into 2 element subsets of the form 1 furthermore any finite boolean ring r is characterized by the degree and coefficients of the characteristic polynomial of a

log lipschitz regularity for sg hyperbolic systems we consider the cauchy problem for linear and quasilinear symmetrizable hyperbolic systems with coefficients depending on time and space not smooth in t and growing at infinity with respect to x we discuss well posedness in weighted sobolev spaces showing that the non lipschitz regularity in t has an influence not only on the loss of derivatives of the solution but also on its behaviour for x we provide examples to prove that the latter phenomenon cannot be avoided

beck s coloring of a commutative ring a commutative ring r can be considered as a simple graph whose vertices are the elements of r and two different elements x and y of r are adjacent if and only if xy 0 beck conjectured that r cl r we give a counterexample where r is a finite local ring with cl r 5 but r 6 we show that if a is a regular noetherian ring with maximal ideals n1 ns such that each a ni is finite then for r a nn11 nnss r cl r finally we give a complete listing of all finite rings r with r 4

differential periodic riccati equations a note on the existence of an infinite number of periodic strong solutions abstract the differential matrix riccati equation with periodic coefficients is considered it is shown by means of a counter example that a sufficient condition for the existence of an infinite number of periodic non negative definite solutions given in a previous paper is not appropriately stated a suitable adjustment of the assumptions allows one to work out rigorous sufficient conditions thanks to these results a necessary and sufficient condition for the existence of the periodic strong solution which relied on the incorrect condition can be shown to be still valid

coloring of commutative rings the purpose of this article is to present the idea of coloring of a commutative ring this idea establishes a connection between graph theory and commutative ring theory which hopefully will turn out to be mutually beneficial for these two branches of mathemathics in this introductory paper we shall mainly be interested in characterizing and discussing the rings which are finitely colorable leaving aside for the moment possible applications to graph theory let r be a commutative ring we consider r as a simple graph whose vertices are the elements of r such that two different elements x and y are adjacent iff xy 0 we let x r denote the chromatic number of the graph i e the minimal number of colors which can be assigned to the elements of r in such a way that every two adjacent elements have different colors a subset c xi x is called a clique provided xixj 0 for all i j if r contains a clique with n elements and every clique has at most y1 elements we say that the clique number of r is n and write clique r n if the sizes of the cliques in r are not bounded we define clique r co we shall show that clique r co actually entails the existence of an infinite clique obviously x r 3 clique r and for general graph g we certainly may have x g clique g however in the case of commutative rings we have not found any example where x r clique r the lack of such counterexamples together with the fact that we have been able to establish the equality x r clique r for certain rather wide classes of rings like reduced and principal ideal rings motivates the following conjecture

on the zero divisor graph of a commutative ring abstract let r be a commutative ring and r be its zero divisor graph in this paper it is shown that for any finite commutative ring r the edge chromatic number of r is equal to the maximum degree of r unless r is a complete graph of odd order in d f anderson a frazier a lauve p s livingston in lecture notes in pure and appl math vol 220 marcel dekker new york 2001 pp 61 72 it has been proved that if r and s are finite reduced rings which are not fields then r s if and only if r s here we generalize this result and prove that if r is a finite reduced ring which is not isomorphic to z 2 z 2 or to z 6 and s is a ring such that r s then r s

the total graph of a commutative ring abstract let r be a commutative ring with nil r its ideal of nilpotent elements z r its set of zero divisors and reg r its set of regular elements in this paper we introduce and investigate the total graph of r denoted by t r it is the undirected graph with all elements of r as vertices and for distinct x y r the vertices x and y are adjacent if and only if x y z r we also study the three induced subgraphs nil r z r and reg r of t r with vertices nil r z r and reg r respectively

rings whose zero divisor graphs have positive genus it is shown that for a fixed positive integer g there are finitely many isomorphism classes of rings whose zero divisor graph has genus g the proof can then be modified to yield an analogous result for nonorientable genus

non commuting graph of a group abstract let g be a non abelian group and let z g be the center of g associate a graph g called non commuting graph of g with g as follows take g z g as the vertices of g and join two distinct vertices x and y whenever x y y x we want to explore how the graph theoretical properties of g can effect on the group theoretical properties of g we conjecture that if g and h are two non abelian finite groups such that g h then g h among other results we show that if g is a finite non abelian nilpotent group and h is a group such that g h and g h then h is nilpotent

periodic strong solutions of periodically time varying matrix riccati equations abstract generalized conditions for the existence of periodic strong solutions in periodically time varying matrix riccati equations are derived first the precise structure of eigenvalues on the unit circle of the transition matrix associated with the hamil tonian matrix and that of the corresponding generalized eigenvectors are clarified in relation to the quasi detectability and quasi stabilizability of the system then such analysis is applied for deriving the conditions for the existence of a unique periodic real symmetric and non negative definite solution in the periodically time varying matrix riccati equation

holder continuity in time for sg hyperbolic systems we investigate well posedness of the cauchy problem for sg hyperbolic systems with non smooth coefficients with respect to time by assuming the coefficients to be holder continuous we show that this low regularity has a considerable influence on the behavior at infinity of the solution as well as on its regularity this leads to well posedness in suitable gelfand shilov classes of functions on rn a simple example shows the sharpness of our results

zero divisor graphs of non commutative rings abstract in a manner analogous to the commutative case the zero divisor graph of a non commutative ring r can be defined as the directed graph r that its vertices are all non zero zero divisors of r in which for any two distinct vertices x and y x y is an edge if and only if x y 0 we investigate the interplay between the ring theoretic properties of r and the graph theoretic properties of r in this paper it is shown that with finitely many exceptions if r is a ring and s is a finite semisimple ring which is not a field and r s then r s for any finite field f and each integer n 2 we prove that if r is a ring and r m n f then r m n f redmond defined the simple undirected graph r obtaining by deleting all directions on the edges in r we classify all ring r whose r is a complete graph a bipartite graph or a tree

modulus of continuity of the coefficients and loss of derivatives in the strictly hyperbolic cauchy problem abstract we deal with the cauchy problem for a strictly hyperbolic second order operator with non regular coefficients in the time variable it is well known that the problem is well posed in l 2 in case of lipschitz continuous coefficients and that the log lipschitz continuity is the natural threshold for the well posedness in sobolev spaces which in this case holds with a loss of derivatives here we prove that any intermediate modulus of continuity between the lipschitz and the log lipschitz one leads to an energy estimate with arbitrary small loss of derivatives we also provide counterexamples to show that the following classification modulus of continuity loss of derivatives is sharp lipschitz no loss intermediate arbitrary small loss log lipschitz finite loss

systems of matrix rational differential equations arising in connection with linear stochastic systems with markovian jumping abstract in this paper a class of systems of matrix nonlinear differential equations containing as particular cases the systems of coupled riccati differential equations arising in connection with control of some linear stochastic systems is considered the system of differential equations considered in this paper are converted in a suitable nonlinear differential equation on a finite dimensional hilbert space adequately choosen this allows us to use the positivity properties of the linear evolution operator defined by the linear differential equations of lyapunov type our aim is to investigate properties of stabilizing and bounded solutions of the considered differential equations and to obtain some conditions ensuring the existence of such solutions conditions providing the existence of a maximal solution minimal solution respectively with respect to some classes of global solutions are presented it is shown that if the coefficients of the equations are periodic functions all these special solutions stabilizing maximal minimal are periodic functions too whenever possible the probabilistic arguments were avoided and so the results proved in the paper appear as results in the field of differential equations with interest in themselves

the diameter of a zero divisor graph let r be a commutative ring and let z r be its set of nonzero zero divisors the set z r makes up the vertices of the corresponding zero divisor graph r with two distinct vertices forming an edge if the product of the two elements is zero the distance between vertices a and b not necessarily distinct from a is the length of the shortest path connecting them and the diameter of the graph diam r is the sup of these distances for a reduced ring r with nonzero zero divisors 1 diam r diam r x diam r x 3 a complete characterization for the possible diameters is given exclusively in terms of the ideals of r a similar characterization is given for diam r and diam r x when r is nonreduced various examples are provided to illustrate the difficulty in dealing with the power series ring over a nonreduced ring

some properties on the tensor product of graphs obtained by monogenic semigroups abstract in das et al 2013 8 a new graph s m on monogenic semigroups s m with zero having elements 0 x x 2 x 3 x n has been recently defined the vertices are the non zero elements x x 2 x 3 x n and for 1 i j n any two distinct vertices x i and x j are adjacent if x i x j 0 in s m as a continuing study in akgunes et al 2014 3 it has been investigated some well known indices first zagreb index second zagreb index randic index geometric arithmetic index atom bond connectivity index wiener index harary index first and second zagreb eccentricity indices eccentric connectivity index the degree distance over s m in the light of above references our main aim in this paper is to extend these studies over s m to the tensor product in detail we will investigate the diameter radius girth maximum and minimum degree chromatic number clique number and domination number for the tensor product of any two not necessarily different graphs s m 1 and s m 2

on commuting graphs of semisimple rings abstract let r be a non commutative ring the commuting graph of r denoted by r is a graph with vertex set r z r and two distinct vertices a and b are adjacent if ab ba in this paper we investigate some properties of r whenever r is a finite semisimple ring for any finite field f we obtain minimum degree maximum degree and clique number of m n f also it is shown that for any two finite semisimple rings r and s if r s then there are commutative semisimple rings r 1 and s 1 and semisimple ring t such that r t r 1 s t s 1 and r 1 s 1

when a zero divisor graph is planar or a complete r partite graph abstract let r be the zero divisor graph of a commutative ring r an interesting question was proposed by anderson frazier lauve and livingston for which finite commutative rings r is r planar we give an answer to this question more precisely we prove that if r is a local ring with at least 33 elements and r then r is not planar we use the set of the associated primes to find the minimal length of a cycle in r also we determine the rings whose zero divisor graphs are complete r partite graphs and show that for any ring r and prime number p p 3 if r is a finite complete p partite graph then z r p 2 r p 3 and r is isomorphic to exactly one of the rings z p 3 z p x y xy y 2 x z p 2 y py y 2 ps where 1 s p

some remarks on the compressed zero divisor graph abstract let r be a commutative ring with 1 0 the zero divisor graph r of r is the undirected graph with vertices the nonzero zero divisors of r and distinct vertices r and s are adjacent if and only if r s 0 the relation on r given by r s if and only if ann r r ann r s is an equivalence relation the compressed zero divisor graph e r of r is the undirected graph with vertices the equivalence classes induced by other than 0 and 1 and distinct vertices r and s are adjacent if and only if r s 0 let r e be the set of equivalence classes for on r then r e is a commutative monoid with multiplication r s r s in this paper we continue our study of the monoid r e and the compressed zero divisor graph e r we consider several equivalence relations on r and their corresponding graph theoretic translations to r we also show that the girth of e r is three if it contains a cycle and determine the structure of e r when it is acyclic and the monoids r e when e r is a star graph

a combinatorial development of fibonacci numbers in graph spectra abstract consider a graph whose vertices are the distinguishable submultisets of a finite multiset such that two vertices r and s are adjacent if and only if r s in this paper the walks on these graphs are linked to square and domino tilings of rectangular and circular boards among the results provided the number of walks of any given length is shown to be a product of generalized fibonacci numbers as an application the eigenvalues are computed for the unrestricted zero divisor graphs of finite commutative rings without nonzero nilpotents

zero divisor graphs of genus one abstract in this paper we find all rings of the forms z p 1 1 z p n n and z n x x m that have genus at most one

graphs associated to co maximal ideals of commutative rings abstract let r be a commutative ring r with 1 in p k sharma s m bhatwadekar a note on graphical representation of rings j algebra 176 1995 124 127 sharma and bhatwadekar defined a graph on r r with vertices as elements of r where two distinct vertices a and b are adjacent if and only if a r b r r let 2 r be the subgraph of r induced by non unit elements in this paper we derive several properties of 2 r we are able to characterize those rings r for which 2 r j r is a forest and those rings r for which 2 r j r is eulerian where j r is the jacobson radical of r we are also able to show that 2 r max r where g is the clique number of a graph g and max r is the set of maximal ideals of r moreover we find all finite rings r such that the genus of 2 r resp r is at most one

laplacian eigenvalues of the zero divisor graph of the ring zn abstract we study the laplacian eigenvalues of the zero divisor graph z n of the ring z n and prove that z p t is laplacian integral for every prime p and positive integer t 2 we also prove that the laplacian spectral radius and the algebraic connectivity of z n for most of the values of n are respectively the largest and the second smallest eigenvalues of the vertex weighted laplacian matrix of a graph which is defined on the set of proper divisors of n the values of n for which algebraic connectivity and vertex connectivity of z n coincide are also characterized

zero divisor graphs of semigroups the zero divisor graph of a commutative semigroup with zero is a graph whose vertices are the nonzero zero divisors of the semigroup with two distinct vertices joined by an edge in case their product in the semigroup is zero we continue the study of this construction and its extension to a simplicial complex

cauchy problems for hyperbolic systems in r n with irregular principal symbol in time and for x rightarrow infty the aim of this paper is to present an approach for the study of well posedness for diagonalizable hyperbolic systems of pseudo differential equations with characteristics which are not lipschitz continuous with respect to both the time variable t locally and the space variables x rn for x we introduce optimal conditions guaranteeing the well posedness in the scale of the weighted sobolev spaces hs1 s2 rn cf introduction with finite or arbitrarily small loss of regularity we give explicit examples for ill posedness of the cauchy problem in the schwartz spaces when the hypotheses on the growth for x fail