Let n(1), . . . , nk be integers greater than one and set [n(i)] = {0, . . . , n(i) - 1}, 1 < i < k. The graph G(n(1), . . . , n(k)) is obtained by letting all the elements of [n(1)] x x [n(k)] to be the vertices and defining distinct vertices (x(1), . . . , x(k)) and (y(1), . . . , y(k)) to be adjacent if and only if gcd(x(i) + y(i), n(i)) = 1 for all 1 < i < k. In this paper, we show that this large class of graphs has just one Cohen-Macaulay member, namely G(2, . . . , 2), and complementation makes infinitely many Cohen-Macaulay graphs. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Let n≥ 2 be an integer. The Grimaldi graph G(n) is defined by taking the elements of the set { 0, … , n-1 } as vertices. Two distinct vertices x and y are adjacent in G(n) if and only if (x+y, n) =1 . In this paper, we examine the Betti numbers of the edge ideals of these graphs and their complements.

For a given finite commutative ring [Formula: see text] with [Formula: see text], one may associate a graph which is called the total graph of [Formula: see text]. This graph has [Formula: see text] as the vertex set and its two distinct vertices [Formula: see text] and [Formula: see text] are adjacent exactly whenever [Formula: see text] is a zero-divisor of [Formula: see text]. In this paper, we give necessary and sufficient conditions for two classes of total graphs to be Cohen–Macaulay.

Let 𝑛 ≥ 2 be an integer. The graph 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, we give some bounds for the Castelnuovo–Mumford regularity of the edge ideals and their powers for .

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.

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.

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.

In this paper, we present existence and uniqueness of special random impulsive differential evolution equations with nonlocal condition in Hilbert spaces. Moreover we study the stability results for the same evolution equations. Existence and uniqueness results are proved using Banach fixed point theorem where as stability results using fixed point approach and semi group theory. Finally we give some applications of the nonlocal impulsive differential equations as well as evolution equations, which shows the importance of our theoretical results.