For a finite group G and an integer r≥ 2 let P_r(G):=|Hom(ℤ^r,G)|/|G|^r, where (ℤ^r,G) is the set of pairwise commuting r-tuples in G. This paper studies rigidity and extremal behavior of the hierarchy {P_r(G)}_r≥2, together with a low-rank representation-theoretic / TQFT counting bridge. The first main direction is cyclic-index rigidity: for groups with an abelian normal subgroup A and cyclic quotient of order ω, under a natural fixed-subgroup hypothesis we prove the exact all-rank formula P_r(G)=1/ω^r+(1-1/ω^r)(|A∩ Z(G)|/|A|)^r-1, which yields gap and rigidity statements for non-abelian abelian extensions of prime index. The second main direction is the class-2 exponent-p world. We develop a symplectic reduction, obtain closed formulas when |G'|=p, and prove a closed all-r hierarchy in the 𝔽_q-Heisenberg family: P_r(G)=q^-2nr∑_k=0^min(n,r)L_n,k(q)∏_i=0^k-1(q^r-q^i). In particular, inside the 𝔽_q-Heisenberg family the pair (P_2(G),P_3(G)) already determines the isoclinism class. Combining the cyclic-index formula with the known sharp upper bound for the multiple commutativity degree gives equality and near-extremal rigidity, including a stability gap near 11/32 for commuting triples. At the low-rank end we also prove explicit class-number formulas for P_3(G) and P_4(G); these recover the simple-count formulas for the untwisted Drinfeld double and the untwisted quantum triple / double-loop-groupoid algebra.

Let G be a finite simple graph. The annihilation number a(G) is an efficiently computable upper bound on the independence number α(G). We develop a sharp matching-number theory for the gap a(G)-α(G). The strongest general theorem is the exact closed form a(G)-α(G)≤ 2μ(G)+1- ⌈√(6 μ(G))⌉ (μ(G)≥ 1), and the bound is attained for every prescribed matching number. We also prove sharp matching-dependent bounds for forests, bipartite graphs, and König-Egerváry graphs, with equality constructions, equality certificates, and equality criteria. Finally, we treat a TxGraffiti output as a machine-conjecture case study. Using annihilating decompositions together with the classical Havel-Hakimi residue inequality res(G)≤ α(G), we give an independent proof of the TxGraffiti annihilation-residue inequality α(G)≥a(G)+res(G)/Δ(G) for every connected graph G of order at least three, show that both hypotheses are necessary, and compare this proof with a recent Caro-Wei approach. We also refine the Caro-Wei annihilation estimate by an explicit nonnegative slack term, identify its equality cases in degree-sequence form, and combine the refinement with our exact matching-number bound to obtain a combined computable bracket for the independence number and a Gupta-residue bound for the annihilation gap.

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.

Let $\alpha(G)$ denote the cardinality of a maximum independent set, while $\mu(G)$ be the size of a maximum matching in $G=\left( V,E\right) $. It is known that if $\alpha(G)+\mu(G)=\left\vert V\right\vert $, then $G$ is a \textit{König-Egerváry graph. If $\alpha (G)+\mu(G)=\left\vert V\right\vert -1$, then $G$ is an $1$-König-Egerváry graph. If $G$ is not a König-Egerváry graph, and there exists a vertex $v\in V$ (an edge $e\in E$) such that $G-v$ ($G-e$) is König-Egerváry, then $G$ is called a vertex (an edge) almost König-Egerváry graph (respectively). In this paper, we characterize all these types of almost König-Egerváry graphs and present interrelationships between them.

It was proved in (Levit and Mandrescu, 2022) that both (V(G), Crown(G)) and (V(G), CritIndep(G)) are augmentoids, established partial augmentation phenomena for the family Ψ(G) of local maximum independent sets, and asked in Problem 5.5 to characterize the graphs whose family Ψ(G) is an augmentoid. We prove that the answer is positive in full generality: for every finite simple graph G, the set system (V(G),Ψ(G)) is an augmentoid. The proof is constructive. If S,T(G), then the explicit choice A=S ∖ N[T], B=T ∖ N[S] satisfies T∪ A(G), S∪ B(G), |T∪ A|=|S∪ B|. As a structural consequence, for every fixed S(G) the map T↦ S∪ T induces a canonical bijection from Ψ(G-N[S]) onto the members of Ψ(G) containing S, and α(G)=|S|+α(G-N[S]). This decomposition also yields explicit formulas for the intersection and the union of all the maximum independent sets extending S, together with counting formulas for the local maximum and maximum independent sets containing S. We also add a short visual guide to the framework CritIndep(G) ⊆ Crown(G)⊆ Psi(G) and end with several natural follow-up problems suggested by the theorem.

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.

Let G be a finite simple graph. An independent set I of G is critical if |I|-|N(I)|≥|J|-|N(J)| for every independent set J of G. A critical independent set is maximum if it has maximum cardinality. The core and the nucleus of G are defined as the intersection of all maximum independent sets and the intersection of all maximum critical independent sets, respectively. In 2019, Jarden, Levit, and Mandrescu posed the problem of characterizing the graphs satisfying core(G)=nucleus(G). In this paper, we provide a complete solution to this problem. Using Larson's independence decomposition, which partitions any graph into a König–Egerváry component L_G an a 2-bicritical component L_G^c, we establish that core(G)=nucleus(G) holds if and only if core (L_G^c)=∅ and no vertex of corona(G) lies in the boundary between L_G and L_G^c. We also show that the same boundary condition is equivalent to the identity diadem(G)=corona(G) ∩ L(G). Several consequences and related structural properties are also derived.

We study formal path expressions for two edge-labeled two-terminal directed acyclic graph families: directed triangulated grid graphs (TGGs) and directed king graphs. The expression is the path polynomial, namely the formal sum of all source-to-target path products in the free noncommutative semiring, and its length is the number of label occurrences in an explicit formula. For TGGs with fixed depth m and variable size n, the backtracking construction has length Om(nm), while decomposition and alternating decomposition give Om(n logm−1 n). We prove that this bound is globally optimal for depths m ≤ 4 by a binomial-language projection. For every fixed depth, we also prove a matching lower bound within a recursively defined balanced intervaldecomposition model that contains the recursive algorithms. For directed king graphs, backtracking is exponential already in depth 2. Geometric decomposition gives Om(nlog2(4m−2)), and a column-transfer interpretation as a fixedwidth algebraic branching program (ABP) improves this to Om(n1+log2 m). Parity-language restrictions give an unrestricted Ω(n2) lower bound, tight for depth 2, and iterated-matrix-multiplication restrictions give product-depth lower bounds. The final part relates path-polynomial factorizations to min-cuts and two-terminal reliability through zero substitutions. MSC Classification: Primary 68R10; Secondary 05C85 , 68Q25 , 68Q45 , 68Q17 , 68W30

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.

We develop a family-based route to unicyclic graphs whose independence polynomials are unimodal but not log-concave. The paper is organized around one flagship statement: for the explicit KL-closure family U_k,r, with r∈{0,1,2} and admissible k, the independence polynomial is unimodal but not log-concave. The proof separates the closure polynomial into a dominant convolution term and a real-rooted correction term. On the non-log-concavity side, we prove symbolically that the penultimate log-concavity inequality fails for every admissible parameter. On the unimodality side, we prove that the main convolution term H_k,r=G_kF_k+r is unimodal with a controlled mode, using a combination of exact coefficient formulas, Ibragimov's strong-unimodality principle, and a residue-class growth argument. Darroch localization and an adjacent-mode bridge lemma then transfer that mode statement to the full KL closure polynomial. This yields an explicit infinite family of unicyclic graphs with unimodal but non-log-concave independence polynomials. In the exact range k≤ 400, we further verify that the penultimate break is unique and determine exact mode formulas for H_k,r, the binomial correction term, and I(U_k,r;x) itself. The paper also places the KL family inside a broader reservoir program involving Galvin, Ramos-Sun, and Bautista-Ramos trees, from which we obtain substantial universal exact theorems for finite ranges.

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.

For a finite group G, we study the higher commuting probabilities, namely the probabilities that r randomly chosen elements of G commute pairwise, together with the corresponding numbers of simultaneous conjugacy classes of commuting r-tuples. We prove an exact dominant asymptotic for the number of homomorphisms from the free abelian group of rank r to G. The exponential base is the maximum order of an abelian subgroup of G, and the leading coefficient is the number of abelian subgroups of that order. As a consequence, the r-th root of the higher commuting probability tends to this maximum abelian-subgroup order divided by the order of G, while the r-th root of the orbit count tends to the maximum abelian-subgroup order itself. We also prove that the associated rank-generating series is rational and has a finite Dirichlet-spectrum expansion supported on abelian subgroup indices. This spectrum yields a finite linear recurrence, a finite-rank Hankel matrix, and an inverse finite-spectrum theorem: the tail of the hierarchy determines the full abelian-index spectrum. For split abelian extensions, we express the dominant base through fixed-subgroup geometry, and for abelian acting quotients, we obtain an exact subgroup-lattice formula. In the cyclic and coprime cases, this gives closed formulas for all spectral coefficients.

A graph is said to be K & odblac;nig-Egerv & aacute;ry if its matching number equals its vertex cover number. The difference between these two graph parameters, the vertex cover number minus the matching number, measures, in some sense, how far a graph is from being a K & odblac;nig-Egerv & aacute;ry graph. Several properties of this difference, called the K & odblac;nig- Egerv & aacute;ry index or K & odblac;nig deficiency, are presented, including some nontrivial structural characterizations. Furthermore, it is shown that various statements involving K & odblac;nig- Egerv & aacute;ry graphs are, in fact, general statements about graphs that can be expressed in terms of their K & odblac;nig-Egerv & aacute;ry indices. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

A graph G is well-covered if all its maximal independent sets are of the same cardinality. Let w:V(G) ⟶ℝ be a weight function. Then G is w-well-covered if all its maximal independent sets are of the same weight. An edge xy ∈ E(G) is relating if there exists S ⊆ V(G) such that both S ∪{x} and S ∪{y} are maximal independent sets. If xy is relating then w(x)=w(y) for every weight function w such that G is w-well-covered. Relating edges are of crucial importance for investigating w-well-covered graphs. The problem whether an edge is relating is NP-complete. We prove that this problem remains NP-complete even for graphs without cycles of length 6. A graph G belongs to the class 𝐖_2 if every two pairwise disjoint independent sets in G are included in two pairwise disjoint maximum independent sets. A vertex v ∈ V(G) is shedding if for every independent set S ⊆ V(G) ∖ N[v] there exists u ∈ N(v) such that S ∪{u} is independent. Shedding vertices play an important role in studying the class 𝐖_2 . Recognizing shedding vertices is co-NP-complete. We prove that this problem is co-NP-complete even for graphs without cycles of length 6.

A set S subset of V is independent in a graph G = (V, E) if no two vertices from S are adjacent. The independence number alpha(G) is the cardinality of a maximum independent set, while mu(G) is the size of a maximum matching in G. If alpha(G)+mu(G) equals the order of G, then G is a K & ouml;nig-Egerv & aacute;ry graph (Deming, 1979; Gavril, 1977; Sterboul, 1979). The number d (G) = max{eAe - eN (A)e : A subset of V} is the critical difference of G (Zhang, 1990) (where N (A) = {v : v E V, N (v) f1A not equal empty set}). It is known that the inequality alpha(G) - mu(G) <= d (G) holds for every graph (Levit and Mandrescu, 2012; Lorentzen, 1966; Schrijver, 2003). one odd cycle. Let ker(G) = boolean AND{S : S is a critical independent set of G}, core(G) be A graph G is (i) unicyclic if it has a unique cycle, (ii) almost bipartite if it has only the intersection of all maximum independent sets, and corona(G) be the union of all maximum independent sets of G. It is known that ker(G) subset of core(G) for every graph (Levit and Mandrescu, 2012), while the equality holds for bipartite graphs (Levit and Mandrescu, 2013), and for unicyclic non-K & ouml;nig-Egerv & aacute;ry graphs (Levit and Mandrescu, 2014). In this paper, we prove that if G is an almost bipartite non-K & ouml;nig-Egerv & aacute;ry graph, then ker(G) = core(G), corona(G) boolean OR N(core(G)) = V(G), and ecorona(G)e + ecore(G)e = 2 alpha(G) + 1. (c) 2025 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

Let α (G) denote the cardinality of a maximum independent set and μ (G) be the size of a maximum matching of a graph G=( V( G) ,E( G) ) . If α (G)+μ (G)=| V( G) | -k , then G is a k -König–Egerváry graph. In particular, if k=0 , then G is a König–Egerváry graph. The corona H∘𝒳 of a graph H and a family of graphs 𝒳={ X_i:1≤ i≤| V(H)| } is obtained by joining each vertex v_i of H to all the vertices of the corresponding graph X_i,i=1,2,...,| V(H)| . In this paper we completely characterize graphs whose coronas are k-König–Egerváry graphs, where k∈{ 0,1} .

An independent set in a graph is a collection of vertices that are not adjacent to each other. The cardinality of the largest independent set in $G$ is represented by $\alpha(G)$. The independence polynomial of a graph $G = (V, E)$ was introduced by Gutman and Harary in 1983 and is defined as \[ I(G;x) = \sum_{k=0}^{\alpha(G)}{s_k}x^{k}={s_0}+{s_1}x+{s_2}x^{2}+...+{s_{\alpha(G)}}x^{\alpha(G)}, \] where $s_k$ represents the number of independent sets in $G$ of size $k$. The conjecture made by Alavi, Malde, Schwenk, and Erd\"os in 1987 stated that the independence polynomials of trees are unimodal, and many researchers believed that this conjecture could be strengthened up to its corresponding log-concave version. However, in our paper, we present evidence that contradicts this assumption by introducing infinite families of trees whose independence polynomials are not log-concave.

An independent set in a graph comprises vertices that are not adjacent to one another, whereas a clique consists of vertices where all pairs are adjacent. For a given graph G, let the following notations be defined: the number of vertices in G is n, the cardinality of a maximum independent set in G is α, the size of the largest clique in G is ω, the cardinality of the intersection of all maximum independent sets in G is ξ, and the number of maximum independent sets in G is sα. As the main finding of this article, we present an upper bound on the number of maximum independent sets as follows: sα≤ω·2n−α−ω+1,ifn−α−ω+1≤α−ξ−1;n−α−ω+1α−ξ+ω·∑k=0α−ξ−1n−α−ω+1k,ifn−α−ω+1≥α−ξ.. As an application of our findings, we explore a series of inequalities that connects the number of longest increasing subsequences with the number of longest decreasing subsequences in a given sequence of integers.

Let α(G) and μ(G) denote the cardinality of a maximum independent set and the size of a maximum matching, respectively, in the graph G= (V,E) . If α(G)+μ(G)= | V | , then G is a Kőnig–Egerváry graph. The number d (G) =max{| A | - | N (A) | :A⊆ V} is the critical difference of the graph G, where N (A) ={ v:v∈ V,N (v) ∩ A≠∅} . Every set B⊆ V satisfying d (G) = | B | - | N (B) | is critical. Let ε (G) = |(G) | and ξ (G) = |core (G) | , where (G) is the intersection of all critical independent sets, and core (G) is the intersection of all maximum independent sets. It is known that (G)⊆ core (G) holds for every graph. Let us define Clearly, ϱ_v (G) = | V | and ϱ_e (G) = | E | for bipartite graphs. Unlike the bipartiteness, the property of being a Kőnig–Egerváry graph is not hereditary. In this paper, we show that ϱ_v (G) = | V | -ξ (G) +ε (G)aaandaaϱ_e (G) ≥| E | -ξ (G) +ε (G) for every Kőnig–Egerváry graph G.