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Let G be a finite group of even order. We give some bounds for the probability p(G) that a randomly chosen element in G has a square root. In particular, we prove that p(G) ≤ 1 - |√|G|/|G|. Moreover, we show that if the Sylow 2-subgroup of G is not a proper normal elementary abelian subgroup of G, then p(G) ≤ 1 - 1/√|G|. Both of these bounds are best possible upper bounds for p(G), depending only on the order of G. © Instytut Matematyczny PAN, 2008  

In this paper the solutions of a two-endpoint boundary value problem is studied and under suitable assumptions the existence and uniqueness of a solution is proved. As a consequence, a condition to guarantee the existence of at least one periodic solution for a class of Liénard equations is presented  

Let δ be a (d-1)-dimensional simplicial complex and let h(δ)=(h0,h1,...,hd) be its h-vector. A recent result of Murai and Terai guarantees that if δ satisfies Serre's condition (Sr), then (h0,h1,...,hr) is an M-vector and hr+hr+1+...+hd is nonnegative. In this article, we extend the result of Murai and Terai by giving r extra necessary conditions. More precisely, we prove that if δ satisfies Serre's condition (Sr), then iihr+i+1ihr+1+...+i+d-rihd, 0≤i≤r≤d, are all nonnegative  

The unit graph corresponding to an associative ring R is the graph obtained by setting all the elements of R to be the vertices and defining distinct vertices x and y to be adjacent if and only if x + y is a unit of R. By a constructive method, we derive necessary and sufficient conditions for unit graphs to be Hamiltonian  

A graph is called weakly perfect if its chromatic number equals its clique number. In this note a new class of weakly perfect graphs is presented and an explicit formula for the chromatic number of such graphs is given  

In this note, we show that under certain assumptions the scalar Riccati differential equation x′=a(t)x+b(t)x 2+c(t) with periodic coefficients admits at least one periodic solution. Also, we give two illustrative examples in order to indicate the validity of the assumptions  

A graph is called weakly perfect if its chromatic number equals its clique number. In this paper a new class of weakly perfect graphs arising from rings are presented and an explicit formula for the chromatic number of such graphs is given. Copyright  

For a given finite commutative ring R with 1≠0, one may associate a graph which is called the total graph of R and it is denoted by T(R). This graph has R as the vertex set and its two distinct vertices x and y are adjacent exactly whenever x+y is a zero-divisor of R. In this note, we prove that T(R) is well-covered if and only if either R is local or 2 is a zero-divisor. © 2021 Elsevier GmbH  

Let R be a finite commutative ring with nonzero identity. The unit graph of R is the graph in which the vertex set is R, and two distinct vertices x and y are adjacent if and only if x + y is a unit in R. In this paper, we determine when these graphs are well-covered, and then, by applying this result, we characterize the unit graphs whose edge rings are Cohen–Macaulay (Gorenstein). This characterization gives us a large class of non-Cohen–Macaulay graphs. © 2022 Taylor & Francis Group, LLC  

For a given finite commutative ring R with 1a0, one may associate a graph which is called the total graph of R. This graph has R as the vertex set and its two distinct vertices x and y are adjacent exactly whenever x + y is a zero-divisor of R. In this paper, we give necessary and sufficient conditions for two classes of total graphs to be Cohen-Macaulay. © 2023 World Scientific Publishing Company  

The unit graph of a ring R with nonzero identity is the graph in which the vertex set is R, and two distinct vertices x and y are adjacent if and only if x + y is a unit in R. In this paper, we derive several necessary conditions for the nonplanarity of the unit graphs of finite commutative rings with nonzero identity, and determine, up to isomorphism, all finite commutative rings with nonzero identity whose unit graphs are toroidal  

In this paper, we characterize complete partite zero-divisor graphs of posets via the ideals of the posets. In particular, for complete bipartite zero-divisor graphs, we give a characterization based on the prime ideals of the posets  

In this paper, we discuss some recent results on graphs attached to rings. In particular, we deal with comaximal graphs, unit graphs, and total graphs. We then define the notion of cototal graph and, using this graph, we characterize the rings which are additively generated by their zero divisors. Finally, we glance at graphs attached to other algebraic structures  

Let R be a finite commutative ring with nonzero identity. The unit graph of R is the graph obtained by setting all the elements of R to be the vertices and defining distinct vertices x and y to be adjacent if and only if x + y is a unit ele¬ment of R. In this paper, a classification of finite commutative rings with nonzero identity in which their unit graphs have domination number less than four is given  

Let R be a ring with nonzero identity. The unit graph of R, denoted by G(R), has its set of vertices equal to the set of all elements of R; distinct vertices x and y are adjacent if and only if x + y is a unit of R. In this article, the basic properties of G(R) are investigated and some characterization results regarding connectedness, chromatic index, diameter, girth, and planarity of G(R)are given. (These terms are defined in Definitions and Remarks 4.1, 5.1, 5.3, 5.9, and 5.13.)  

Let R be a finite commutative ring with nonzero identity and denote its Jacobson radical by J(R). The Jacobson graph of R is the graph in which the vertex set is RJ(R), and two distinct vertices x and y are adjacent if and only if 1-xy is not a unit in R. In this paper, the nonorientable genus of some Jacobson graphs is either computed or estimated by a lower bound. As an application, the rings R with projective Jacobson graphs are classified  

Let R be a finite commutative ring with nonzero identity and denote its Jacobson radical by J(R). The Jacobson graph of R is the graph in which the vertex set is R J(R) , and two distinct vertices x and y are adjacent if and only if 1 - xy is not a unit in R. In this paper, up to isomorphism, we classify the rings R whose Jacobson graphs are toroidal. © 2016, Malaysian Mathematical Sciences Society and Universiti Sains Malaysia  

Let n be a positive integer and let Sn be the set of all nonnegative integers less than n which are relatively prime to n. In this paper, we discuss structural properties of circulant graphs generated by the Sn′s and their complements. In particular, we characterize when these graphs are well-covered, Cohen–Macaulay, Buchsbaum or Gorenstein. © 2020, Springer Science+Business Media, LLC, part of Springer Nature