We characterize the graphs which are simultaneously α-critical and members of the class 𝐖_p. The characterization is stated in three equivalent languages. In the graph itself, such a graph is a well-covered graph whose codimension-one localization fibers all have size at least p and whose edges are exactly covered by the cliques induced by those fibers. In the independence complex, it is a pure flag complex in which every ridge has degree at least p and every missing edge is generated by the link of a ridge. In the complement, it is a K_r+1-saturated graph, where r=α(G), all maximal cliques have size r, and the minimum (r-1)-clique-codegree is at least p. This gives an exact formula for the largest p for which a well-covered graph belongs to 𝐖_p. We make this complement correspondence explicit, record saturation-theoretic consequences including dense-complement rigidity and p-sensitive edge and order bounds, and give a family of sharp examples showing that the local sufficient condition from the recent work of Hoang, Levit and Mandrescu is not necessary outside the locally triangle-free setting, for all p≥2.

We completely determine the asymptotic depth, equivalently, the asymptotic projective dimension of a chain of edge ideals that is invariant under the action of the monoid Inc of increasing functions on the positive integers. Our results and their proofs also reveal surprising combinatorial and topological properties of corresponding graphs and their independence complexes. In particular, we are able to determine the asymptotic behavior of all reduced homology groups of these independence complexes. (c) 2026 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

We establish new characterizations of graphs belonging to the Wp class. In addition, we characterize locally triangle-free alpha-critical graphs in this class. As a consequence, our results yield a partial answer to a question raised by Plummer [M.D. Plummer, Well-covered graphs: A survey, Quaest. Math. 16 (1993) 253-287] in the case p = 2.

For a positive integer p , a graph $G$ belongs to class W p if |V(G)| > p and any p disjoint independent sets are contained in p disjoint maximum independent sets. In this paper, we establish some inequalities for the coefficients of the independence polynomial of a W p graph.It is then proven that the clique corona graph G ο K p of a graph G and a complete graph K p belongs to W p . As an application, weexamine the unimodality problem for the independence polynomial of G ο K p .In addition we prove the unimodality of the independence polynomial ofS n ο K p , where S n is a star graph.

Let G be a graph of order n. For a positive integer p, G is said to be a Wp graph if n >= p and every p pairwise disjoint independent sets of G are contained within p pairwise disjoint maximum independent sets. In this paper, we establish that every connected Wp graph G is p-quasi-regularizable if and only if n >= (p + 1) & centerdot; alpha, where alpha is the independence number of G and p =/ 2. This finding ensures that the independence polynomial of a connected Wp graph G is log-concave whenever (p + 1) & centerdot; alpha <= n <= p & centerdot; alpha + 2 root p & centerdot; alpha + p and alpha 2 <= p, or p & centerdot; alpha + 2 root p & centerdot; alpha + p 2+1)& centerdot;p+(alpha-1)2 alpha+1 <= p. Moreover, the clique corona graph G degrees Kp serves as an example of the Wp graph class. We further demonstrate that the independence polynomial of G degrees Kpis always log-concave for sufficiently large p. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Let H = {H-v : v is an element of V (G)} be a family of nonempty graphs indexed by the vertex set of a graph G. The corona graph G degrees H of G and H is the disjoint union of G and H-v, v is an element of V (G), with additional edges joining each vertex v is an element of V (G) to all the vertices of H-v. In this paper, we are deeply concerned in investigating the (induced) matching number of the graph G degrees H. It consequently gives an explicit formula to compute the Castelnuovo-Mumford regularity of edge ideal of this graph. Moreover, we propose a close relationship between independence polynomial of corona graph and h-polynomial of its independence simplicial complexes. Thereby, the formula of h-polynomial is established and some results related to the unimodality and real-rootedness are presented.

In this paper, our research primarily focuses on f-symmetric and unimodal properties of independence polynomials I(G∘ (K_p∪ K_q);x) . More precisely, we prove that the independence polynomial I(G∘ (K_p∪ K_q);x) is unimodal for every graph G, whenever the positive integers p and q are large enough. In addition, we establish several inequalities involving the coefficients of I(G∘ (K_p∪ K_q);x) for arbitrary p and q.

If for any $k$ the $k$-th coefficient of a polynomial $I(G;x)$ is equal to the number of stable sets of cardinality $k$ in the graph $G$, then it is called the independence polynomial of $G$ (Gutman and Harary, 1983). Alavi, Malde, Schwenk and Erdos (1987) conjectured that $I(G;x)$ is unimodal, whenever $G$ is a forest, while Brown, Dilcher and Nowakowski (2000) conjectured that $I(G;x)$ is unimodal for any well-covered graph G. Michael and Traves (2003) showed that the assertion is false for well-covered graphs with $a(G)$ > 3 ($a(G)$ is the size of a maximum stable set of the graph $G$), while for very well-covered graphs the conjecture is still open. In this paper we give support to both conjectures by demonstrating that if $a(G)$ < 4, or $G$ belongs to ${K_{1,n}, P_{n}: n > 0}$, then $I(G*;x)$ is log-concave, and, hence, unimodal (where $G*$ is the very well-covered graph obtained from $G$ by appending a single pendant edge to each vertex).

In this paper, our research primarily focuses on f -symmetric and unimodal properties of independence polynomials I (G circle (K-p boolean OR K-q); x). More precisely, we prove that the independence polynomial I (G circle (K-p boolean OR K-q)); x) is unimodal for every graph G, whenever the positive integers p and q are large enough. In addition, we establish several inequalities involving the coefficients of I (G circle (K-p boolean OR K-q)); x) for arbitrary p and q.

We study chains of nonzero edge ideals that are invariant under the action of the monoid Inc of increasing functions on the positive integers. We prove that the sequence of Castelnuovo–Mumford regularity of ideals in such a chain is eventually constant with limit either 2 or 3, and we determine explicitly when the constancy behavior sets in. This provides further evidence to a conjecture on the asymptotic linearity of the regularity of Inc -invariant chains of homogeneous ideals. The proofs reveal unexpected combinatorial properties of Inc -invariant chains of edge ideals.

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 $$\mathcal H = \{H_v: v \in V (G)\}$$ be a family of nonempty graphs indexed by the vertex set of a graph G. The corona $$G\circ \mathcal H$$ of G and $$\mathcal H$$ is the disjoint union of G and $$H_v$$ , $$v \in V (G)$$ , with additional edges joining each vertex $$v \in V (G)$$ to all the vertices of $$H_v$$ . In this paper, we show that the corona graph $$G\circ \mathcal H$$ is Cohen–Macaulay if and only if $$G\circ \mathcal H$$ is a clique corona graph, i.e., all graphs $$H_v$$ in $$\mathcal H$$ are complete graphs. In addition, if I denotes the edge ideal of the clique corona graph $$G\circ \mathcal H$$ , we prove that the Castelnuovo–Mumford regularity of R/I is equal to the induced matching number of $$G\circ \mathcal H$$ .

We give an explicit formula for the Hilbert–Poincaré series of the parity binomial edge ideal of a complete graph  $$K_{n}$$ or equivalently for the ideal generated by all $$2\times 2$$ -permanents of a $$2\times n$$ -matrix. It follows that the depth and Castelnuovo–Mumford regularity of these ideals are independent of n.

We show that Cohen-Macaulay and (S 2 ) properties are equivalent for the second power of an edge ideal. We give an example of a Gorenstein squarefree monomial ideal I such that S / I 2 satisfies the Serre condition (S 2 ), but is not Cohen-Macaulay.

We study the equality of the extremal Betti numbers of the binomial edge ideal $J_G$ and those of its initial ideal ${\rm in}(J_G)$ of a closed graph $G$. We prove that in some cases there is an unique extremal Betti number for ${\rm in}(J_G)$ and as a consequence there is an unique extremal Betti number for $J_G$ and these extremal Betti numbers are equal

We characterize all graphs for which the saturation of the second power of their edge ideals is Cohen–Macaulay.

Let $I(G)$ be the edge ideal of a simple graph $G$. In this paper, we will give sufficient and necessary combinatorial conditions of $G$ in which the second symbolic and ordinary power of its edge ideal are Cohen-Macaulay (resp. Buchsbaum, generalized Cohen-Macaulay). As an application of our results, we will classify all bipartite graphs in which the second (symbolic) powers are Cohen-Macaulay (resp. Buchsbaum, generalized Cohen-Macaulay).