harmonic analysis on nilpotent groups and singular integrals iii fractional integration along manifolds in this paper we continue the analysis of integral operators on nilpotent lie groups with kernels supported on submanifolds and or containing an oscillatory factor of polynomial type in our previous articles 13 143 we have essentially considered singular integral operators i e those defined by kernels having the critical degree of homogeneity here we focus our attention on kernels having a higher degree of homogeneity so that locally they have a finite mass but do not have enough decay at infinity more precisely if q is the homogeneous dimension of the group g which in this article we always assume endowed with automorphic dilations we consider kernels k having the same homogeneity as 1x1 q l a where 1x1 denotes a homogeneous gauge and re a 0 these kernels are assumed to be supported on analytic homogeneous submanifolds v we call them fractional integration kernels along v we discuss three interrelated problems concerning fractional integration kernels

weak type estimates for a singular convolution operator on the heisenberg group on the heisenberg group ehn with coordinates z t e cn x r define the distribution k z t l z 6 t where l z is a homogeneous distribution on cn of degree 2n smooth away from the origin and 3 t is the dirac mass in the t variable we prove that the operator given by convolution with k maps hl h n to weak l1 hn

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

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

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

harmonic analysis on nilpotent groups and singular integrals ii singular kernels supported on submanifolds on etudie des operateurs de convolution singuliers sur des groupes de lie nilpotents generaux et plus specifiquement des operateurs dont les noyaux ont pour supports des varietes de faible dimension et qui peuvent aussi contenir des facteurs polynomiaux exponentiels oscillatoires

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

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

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

a note on automorphisms of the zero divisor graph of upper triangular matrices abstract let f q be a finite field with q elements n 3 a positive integer t n q the set of all n n upper triangular matrices over f q in 13 the zero divisor graph of t n q written as t is defined to be a graph with all nonzero zero divisors in t n q as vertices and there is a directed edge from a vertex x to a vertex y if and only if x y 0 the subgraph of t induced by all rank one matrices in t n q is denoted by r wong et al 2014 in 13 determined the automorphisms of r and left the automorphisms of t unsolved in this note we solve this problem

network landscape from a brownian particle s perspective given a complex biological or social network how many clusters should it be decomposed into we define the distance d i j from node i to node j as the average number of steps a brownian particle takes to reach j from i node j is a global attractor of i if d i j l d i k for any k of the graph it is a local attractor of i if j ensuremath in e i the set of nearest neighbors of i and d i j l d i l for any l ensuremath in e i based on the intuition that each node should have a high probability to be in the same community as its global local attractor on the global local scale we present a simple method to uncover a network s community structure this method is applied to several real networks and some discussion on its possible extensions is made

inversion of the radon transform on the laguerre hypergroup by using generalized wavelets abstract we consider the radon transform r 0 on the laguerre hypergroup k 0 r we characterize a space of infinitely differentiable and rapidly decreasing functions together with their derivatives such that r is a bijection from this space onto itself we establish an inversion formula and a plancherel theorem for the operator r finally by using the continuous wavelet transform on the laguerre hypergroup k we deduce another expression for the inverse r 1 of the operator r

the annihilating ideal graph of commutative semigroups abstract in this paper we associate an undirected graph ag s the annihilating ideal graph to a commutative semigroup s this graph has vertex set a s a s 0 where a s is the set of proper ideals of s with nonzero annihilator two distinct vertices i j a s are defined to be adjacent in ag s if and only if i j 0 the zero ideal conditions are given to ensure a finite graph semigroups for which each nonzero proper ideal of s is an element of a s are characterized connections are drawn between ag s and s the well known zero divisor graph and the connectivity diameter and girth of ag s are described semigroups s for which ag s is a complete or star graph are characterized finally it is proven that the chromatic number is equal to the clique number of the annihilating ideal graph for each reduced semigroup and null semigroup upper and lower bounds for ag s are given for a general commutative semigroup

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

weighted weak 1 1 and weighted estimates for oscillating kernels weak type 1 1 and strong type p p inequalities are proved for operators defined by oscillating kernels the techniques are sufficiently general to derive versions of these inequalities using weighted norms 0 introduction given a positive real number a o a 7 1 define the oscillating kernel ka by ka x 1 x leilxla and consider the convolution operator ka f in an earlier paper 2 we studied the boundedness properties of such operators on weighted lp spaces 1 1 we define l p f 1l p w a e f x lpw x dx r and say f e lp r if llfllp w 1 the multiplier operator t associated to ka is defixled by t d 0 1 l e 1tl f where 0 is a c function defined by i 0 4 0 if 1 when p 0

the radon transform on the heisenberg group and the transversal radon transform abstract the radon transform on the heisenberg group was introduced by r strichartz we regard it as a particular case of a more general transversal radon transform that integrates functions on r m over hyperplanes meeting the last coordinate axis the paper contains new boundedness results and explicit inversion formulas for both transforms of l p functions in the full range of the parameter p we also show that these transforms are isomorphisms of the corresponding semyanistyi lizorkin spaces of smooth functions in the framework of these spaces we obtain inversion formulas which are pointwise analogues of the corresponding formulas by r strichartz

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

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

comaximal graph of commutative rings abstract let r be a commutative ring with identity let r be a graph with vertices as elements of r where two distinct vertices a and b are adjacent if and only if r a r b r in this paper we consider a subgraph 2 r of r which consists of non unit elements we look at the connectedness and the diameter of this graph we completely characterize the diameter of the graph 2 r j r in addition it is shown that for two finite semi local rings r and s if r is reduced then r s if and only if r s

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

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

singular and maximal radon transforms analysis and geometry part 2 geometric theory 8 curvature introduction 8 1 three notions of curvature 8 2 theorems 8 3 examples 9 curvature some details 9 1 the exponential representation 9 2 diffeomorphism invariance 9 3 curvature condition cy 9 4 two lemmas 9 5 double fibration formulation 10 equivalence of curvature conditions 10 1 invariant submanifolds and deficient lie algebras 10 2 vanishing jacobians 10 3 construction of invariant submanifolds

an fio calculus for marine seismic imaging ii sobolev estimates we establish sharp l 2 sobolev estimates for classes of pseudodifferential operators with singular symbols guillemin and uhlmann duke math j 48 251 267 1981 melrose and uhlmann commun pure appl math 32 483 519 1979 whose non pseudodifferential fourier integral operator parts exhibit two sided fold singularities the operators considered include both singular integral operators along curves in mathbb r 2 with simple inflection points and normal operators arising in linearized seismic imaging in the presence of fold caustics felea comm pde 30 1717 1740 2005 felea and greenleaf comm pde 33 45 77 2008 nolan siam j appl math 61 659 672 2000

two parameter maximal functions in the heisenberg group soit h 1 le groupe de heisenberg a 3 dimensions constitue des matrices triangulaires inferieures g x y z on etudie des operateurs maximaux sur h 1 associes a la famille a 2 parametres des dilatations d e x y z sx ey z et en determinant leurs proprietes de l p limitation

hilbert integrals singular integrals and radon transforms i on introduit deux nouvelles classes d operateurs on etudie leur relation et on montre comment on peut les appliquer a l etude des problemes aux valeurs limites

lp harmonic analysis and radon transforms on the heisenberg group in this paper harmonic analysis in the heisenberg group heisn cn r with group law z t z t z z t t 12im z z is taken to mean the joint spectral theory of the operators l heisenberg laplacian and t t the spectral decomposition of e l2 is given explicitly as e 0 e d where e 1 k 0 1 2 and e z t n 2 n 1 n 2k n 1 exp ie tn 2k exp z 24 n 2k lkn 1 z 22 n 2k where lkn 1 denotes the laguerre polynomial the eigenvalues are it e e n 2k e and l k e k e this decomposition is essentially well known but has not previously been described in this light it is also the decomposition into irreducibles of the representation of the heisenberg motion group semi direct product of heisn and u n on l2 heisn the main result is a summability in lp of the decomposition for e lp 1 p limr 1 rk 0 ed for p 1 only a weaker substitute is available and for p the decomposition is false there is also a plancherel formula for l2 2 2 k e n 2k 0 k e z 0 2dzd a number of explicit examples of the decomposition are computed including part of the schrodinger propagator eisl the heisenberg radon transform is defined by r z t n z t w 0 dw this operator was studied by geller and stein it is shown to be bounded from lp to lq if and only if p 2n 2 2n 1 and q 2n 2 some generalizations of these results to the free step two nilpotent lie groups are given but the results are not as complete

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

optimal routing strategy based on the minimum information path abstract in this paper we propose a new routing strategy based on the minimum information path named the optimal routing or strategy to improve the transportation capacity of scale free networks we define the average routing centrality degree of the node to analyze the traffic load on nodes of different degree we analyze the transportation capacity by using the critical values of r c the average packet travel time and the average path length both theoretical and experimental results show that the capacity of the network under our strategy will be maximized when the packet delivery rate of the node is directly proportional to the degree

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

hilbert integrals singular integrals and radon transforms ii etude des transformations de radon singulieres et des integrales de hilbert appliquees aux problemes aux valeurs limites probleme de neumann

co maximal graph of non commutative rings let r be a ring not necessarily commutative with 1 following sharma and bhatwadekar p k sharma s m bhatwadekar a note on graphical representation of rings j algebra 176 1995 124 127 we define a graph on r r with vertices as elements of non units of r where two distinct vertices a and b are adjacent if and only if ra rb r in this paper we investigate the behavior of r we are able to prove that if r is left artinian then r j r is connected and if r j r is a forest then r j r is a star graph where j r is the jacobson radical of r for any finite field fq we obtain the minimal degree the maximal degree the connectivity the clique number and the chromatic number of mn fq finally for any finite field and any integer n 2 we prove that if r is a ring with identity and r mn f then r mn f we also prove that if r and s are two finite commutative rings with identity and n m 2 such that mn r mm s then n m and r s provided that r is reduced

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

a new bound of radius with irregularity index in this paper we use a technique introduced in the paper p dankelmann r c entringer average distance minimum degree and spanning trees j graph theory 33 2000 1 13 to obtain a strengthening of an old classical theorem by erdos et al p erdos j pach r pollack z tuza radius diameter and minimum degree j combin theory b 47 1989 73 79 on radius and minimum degree to be more detailed we will prove that if g is a connected graph of order n with the minimum degree d then the radius g does not email protected 1 1 where t is the irregularity index that is the number of distinct terms of the degree sequence of g which has been recently defined in the paper s mukwembi a note on diameter and the degree sequence of a graph appl math lett 25 2012 175 178 we claim that our result represent the tightest bound that ever been obtained until now

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

detecting unknown paths on complex networks through random walks abstract in this article we investigate the problem of detecting unknown paths on complex networks through random walks to detect a given path on a network a random walker should pass through the path from its initial node to its terminal node in turn we calculate probability t that a random walker detects a given path on a connected network in t steps when it starts out from source node s we propose an iteration formula for calculating t generating function of t is also derived major factors affecting t such as walking time t path length l starting point of the walker structure of the path and topological structure of the underlying network are further discussed among these factors two most outstanding ones are walking time t and path length l on the one hand t increases as t increases and 1 which shows that the longer the walking time the higher the chance of detecting a given path and the walker will discover the path sooner or later so long as it keeps wandering on the network on the other hand t drops substantially as path length l increases which shows that the longer the path the lower the chance for the walker to find it in a given time apart from path length path structure also has obvious effect on t starting point of the walker has only minor influence on t but topological structure of the underlying network has strong influence on t simulations confirm our analytic results

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

random walks and diameter of finite scale free networks abstract dynamical scalings for the end to end distance r e e and the number of distinct visited nodes n v of random walks rws on finite scale free networks sfns are studied numerically r e e shows the dynamical scaling behavior r e e l t l n g t l z where l is the average minimum distance between all possible pairs of nodes in the network n is the number of nodes is the degree exponent of the sfn and t is the step number of rws especially r e e l t in the limit t satisfies the relation r e e l d where d is the diameter of network with d l ln n for 3 or d l ln ln n for 3 based on the scaling relation r e e we also find that the scaling behavior of the diameter of networks can be measured very efficiently by using rws

diffusive capture process on complex networks we study the dynamical properties of a diffusing lamb captured by a diffusing lion on the complex networks with various sizes of n we find that the lifetime t of a lamb scales as t approximately n and the survival probability s n infinity t becomes finite on scale free networks with degree exponent gamma 3 however s n t for gamma 3 has a long living tail on tree structured scale free networks and decays exponentially on looped scale free networks this suggests that the second moment of degree distribution k2 is the relevant factor for the dynamical properties in the diffusive capture process we numerically find that the normalized number of capture events at a node with degree k n k decreases as n k approximately k sigma when gamma 3 n k still increases anomalously for k approximately kmax where kmax is the maximum value of k of given networks with size n we analytically show that n k satisfies the relation n k approximately k2 p k for any degree distribution p k and the total number of capture events ntot is proportional to k2 which causes the gamma dependent behavior of s n t and t

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

harmonic analysis on nilpotent groups and singular integrals i oscillatory integrals abstract this paper is devoted to the study of the operator given by where k is a standard calderon zygmund kernel and p is a real polynomial on r n r n we show t is bounded on lp to itself when 1

estimates for singular convolution operators on the heisenberg group soit h n le groupe de heisenberg l operateur a co c h n donne par af f k s etend a un operateur borne de l p dans l p pour 1 p les operateurs ae co c donnes par aef f ke s etendent aux operateurs bornes sur l p pour 1 p avec des normes independantes de e

singular integrals related to the radon transform and boundary value problems abstract two classes of integral operators are introduced that combine features of calderon zygmund singular integrals the hilbert integral and the radon transform examples and applications arising from integral geometry and several complex variables are discussed

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