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In this paper, we deal with very well-covered graphs. We first describe the structure of these kinds of graphs based on the structure of Cohen–Macaulay very well-covered graphs. As an application, we analyze the structure of local cohomology of the residue rings by the edge ideals of very well-covered graphs. Also, we give different formulas of regularity and depth of these rings from known ones and we finally treat the CMt property. © 2022 Elsevier Inc  

A very well-covered graph is a well-covered graph without isolated vertices such that the height of its edge ideal is half of the number of vertices. In this survey article, we gather together most of the old and new results on the edge and cover ideals of these graphs. © 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG  

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  

Let R be a finite commutative ring with nonzero identity. The unitary Cayley graph of R is the graph obtained by letting 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 element of R. In this paper, we characterize all unitary Cayley graphs with Roman domination number at most four. © 2022 The Author(s). Published with license by 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  

Let n≥2 be an integer. The graph G(n) is obtained by letting all the elements of {0,…,n−1} to be the vertices and defining distinct vertices x and y to be adjacent if and only if gcd⁡(x+y,n)=1. In this paper, well-coveredness, Cohen–Macaulayness, vertex-decomposability and Gorensteinness of these graphs and their complements are characterized. These characterizations provide large classes of Cohen–Macaulay and non Cohen–Macaulay graphs. © 2021 Elsevier B.V  

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  

In this note, we show that under certain assumptions the matrix Riccati differential equation X′ = A(t)X + XB(t)X + C(t) with periodic coeffi cients admits at least one periodic solution. Also, we give an illustrative example in order to indicate the validity of the assumptions and the novelty of our result. © 2021, Tsing Hua University. All rights reserved  

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  

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  

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 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. 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 K{double-struck} be a field, and let R = K{double-struck}[x1,.., xn] be the polynomial ring over K{double-struck} in n indeterminates x1,.., xn. Let G be a graph with vertex-set {x1,.., xn}, and let J be the cover ideal of G in R. For a given positive integer k, we denote the kth symbolic power and the kth bracket power of J by J(k) and J[k], respectively. In this paper, we give necessary and sufficient conditions for R/Jk, R/J (k), and R/J [k] to be Cohen-Macaulay. We also study the limit behavior of the depths of these rings  

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 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  

In this paper, we deal with zero-divisor graphs of posets. We prove that the diameter of such a graph is either 1, 2 or 3 while its girth is either 3, 4 or ∞. We also characterize zero-divisor graphs of posets in terms of their diameter and girth  

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