Papers

Let $L(\lambda)$ be a simple highest weight module of a classical Lie algebra $\mathfrak{g}$ with highest weight $\lambda-\rho$, where $\rho$ is half the sum of positive roots. Joseph proved that the associated variety of the annihilator ideal of $L(\lambda)$ (also called the annihilator variety) is the Zariski closure of a nilpotent orbit in $\mathfrak{g}^*$. Recently, Bai–Ma–Wang introduced a partition algorithm to describe this corresponding nilpotent orbit for a given highest weight module $L(\lambda)$. In this paper, we present a new direct proof of Bai–Ma–Wang's partition algorithm using Sommers duality.

In this work, we settle a problem that dates back to the early 2000s. We provide a complete characterization of special matchings of arbitrary Bruhat intervals in Coxeter groups of type $A$ and apply this result to prove a conjecture of Brenti from 2003 concerning the computation of Kazhdan-Lusztig $R$-polynomials via special matchings. This yields new evidence in support of the Combinatorial Invariance Conjecture.

The process of deriving the Desarguesian plane $PG(2,q^2)$ to get the Hall plane is well known, and the problem of when a conic in $PG(2,q^2)$ inherits to an arc in the Hall plane has been solved. In this article we look at the generalisation of replacing an André net of $PG(2,q^t)$, $t\geq 3$ to construct an André plane of order $q^t$. This article looks at the case where $q$ is odd and $t$ is prime, and determines when a conic in $PG(2,q^t)$ inherits to an arc in an André plane. Further, the number of arcs in an André plane that are inherited in this way is enumerated.

Given a graph $G$, let $\chi(G)$ denote the chromatic number of $G$. For $k\in \mathbb{N}$, a graph $G$ is $k$-$vertex$-$critical$ if $\chi(G)=k$ and $\chi(G-v)< k$ for all $v\in V(G)$. A recent problem of Beaton and Cameron [TCS 1042 (2025) 115234] asks for which graphs $H$ of order five, are there finitely many $k$-vertex-critical (co-gem, $H$)-free graphs, for all $k\in \mathbb{N}$? Here we identify three distinct graphs on five vertices that yield an affirmative answer to this problem. More precisely, we show that for each $k\in \mathbb{N}$, there are finitely many $k$-vertex-critical (co-gem, $H$)-free graphs, where $H\in \{$paraglider, dart, house$\}$, by analysing the structure of such graphs. Our results together with a result of Couturier et al. [Algorithmica 71:1 (2015) 21–35] imply that for each $k\in \mathbb{N}$, there is a polynomial-time certifying algorithm for $k$-COLORING of (co-gem, $H$)-free graphs, where $H\in \{$paraglider, dart, house$\}$.

We study discrete analogues of classical spectral geometric inequalities and extremal eigenvalue problems on graphs. The classical Krahn–Szegő inequality states that, among bounded open subsets of $\mathbb{R}^n$ with fixed volume, the minimum of $\lambda_2(\Omega)$ is attained by the union of two congruent balls. Firstly, we establish a Krahn–Szegő type inequality for trees. For trees with a fixed number of interior vertices and boundary leaves, we completely characterize the extremal structures that minimize the second Dirichlet eigenvalue. Secondly, we develop a nodal domain method for adjacency matrices. By proving an adjacency version of the nodal domain theorem for graphs, we obtain upper bounds for the second largest adjacency eigenvalue $\rho_2(G)$ of $G$ in given graph classes. These bounds imply some previous results. Finally, we settle the Aouchiche–Hansen conjecture (2010) on the second largest eigenvalue with given number of edges and clique number. We prove that for connected graphs $G$ of odd order $n \geq 5$, $|\rho_2| \cdot \omega \leq m-2$, with equality if and only if $G$ consists of two complete graphs of orders $\frac{n+1}{2}$ and $\frac{n-1}{2}$ joined by an edge or a path. For even $n \geq 2$, the quantity $|\rho_2| \cdot \omega - m$ is maximized exactly when $G$ is obtained by adding one edge between the two copies of $K_{n/2}$ by an edge. The core of the methods developed in this paper is to regard a connected graph as an internally disconnected graph with Dirichlet boundary condition. This perspective allows us to transfer nodal domain techniques from continuous spectral geometry to discrete settings and to obtain sharp extremal characterizations across diverse graph classes.

When $S$ is a finite set of positive integers, we can consider classical Subtraction Nim with $S$ as the set of removable numbers. Even when $S$ consists of three elements, many questions remain unanswered. For example, we do not have a period formula of the Nim value. In this paper, we generalize $S$ to be a finite set of positive real numbers. We found that in some regions, we can give concrete formulae for the period and the Nim value function. In particular when $S$ consists of three elements, we found sufficient conditions for the Nim value function to be purely periodic with the period which is equal to the sum of two of elements of $S$. To be more precise, let $S = {a,b,c}$ with $0 < a < b < c$, then for example when $a \leq b \leq 2a$ with $a+b \geq c$, the Nim value function is purely periodic with a period $a+c$. There are much more regions with precise period formulae. We have also some generalizations for the cases $|S| \geq 4$. Even when $S$ consists of integers, these results seem to be new.

For every fixed $t\ge 1$, we prove a parity analogue of the mighty Erdős-Hajnal property for $t$-intersecting curves in the plane. Let $\mathcal B$ be a set of blue curves and $\mathcal G$ a set of green curves in the plane such that $\mathcal B\cup\mathcal G$ is a collection of $t$-intersecting curves in general position. We show that there exist subfamilies $\mathcal B'\subseteq\mathcal B$ and $\mathcal G'\subseteq\mathcal G$ such that $|\mathcal B'|\geq \varepsilon|\mathcal B|$ and $|\mathcal G'|\geq \varepsilon|\mathcal G|$, where $\varepsilon>0$ depends only on $t$, such that either every pair in $\mathcal B'\times\mathcal G'$ intersects an even number of times or every such pair intersects an odd number of times. For $t=1$, this recovers the theorem of Fox, Pach, and Suk for pseudo-segments. As an application, we show that every $n$-vertex topological graph with edges forming a $t$-intersecting family and with no $k$ edges that pairwise cross an odd number of times has at most $n(\log n)^{O_t(\log k)}$ edges.

Let $\lambda_i(G)$ denote the $i$-th largest eigenvalue of adjacency matrix of a graph $G$. Gerschgorin's Theorem indicates $\lambda_1(G)$ belongs to the largest disk, i.e., $\lambda_1(G)\le\Delta_1(G)$, where $\Delta_i(G)$ is the $i$-th largest degree of $G$. We show that $\lambda_2(G)$ lies in the second largest disk. That is, in detail, $$\lambda_2(G)<\Delta_2(G)-\frac{1}{n^2}.$$ A classical theorem proved by Hong [Linear Algebra Appl. 1988] states that $\lambda_1(G)\le\sqrt{2m-n+1}$ for a connected graph $G$ with $n$ vertices and $m$ edges, where the equality holds if and only if $G$ is a star $S_n$ or a complete graph $K_n$. We give a refinement of Hong's theorem by showing $$\lambda_1(G)<\sqrt{2m-n}$$ for any connected graph $G\not\in\left\{S_n,S^1_{n-1},K_n,K^1_{n-1}\right\}$. Based on this improved upper bound of $\lambda_1(G)$, for a connected graph $G$ with $n$ vertices and $m$ edges, we are able to prove a sharp upper bound of $\lambda_2(G)$ that $$\lambda_2(G)\le\sqrt{m-\frac{n}{2}-\frac{1}{2}},$$ except $G$ is obtained from two disjoint $S_\frac{n}{2}$ by adding an edge between a pendant vertex of each star. Moreover, we provide a complete characterization to extremal graphs attaining the equality.

In this paper, we demonstrate the effects on the second largest eigenvalue $\lambda_2(G)$ of a connected graph $G$ after edge addition or deletion. In 1989, Chung, Graham and Wilson showed $\max\{|\lambda_2|,|\lambda_n|\}>\Omega(n)$ for dense $K_{r+1}$-free graphs of order $n$, giving spectral comprehension of existence of large clique or independent set, respect to Ramsey theory. Applying the results of effects on $\lambda_2$ after edge operations, we determine the maximum value of $\lambda_2$ among all $K_{r+1}$-free connected graphs with given order, and completely characterize the extremal graphs. Moreover, for arbitrary given graph $F$, we investigates the maximum second largest $\lambda_2(G)$ among $F$-free connected graphs of order $n$. Let $\rho^*(n,F)$ be the maximum spectral radius of $F$-free graphs on $n\ge n_F$ vertices, and $G^*(n,F)$ be a graph with its spectral radius $\rho\big(G^*(n,F)\big)=\rho^*(n,F)$. We prove that, for an $F$-free connected graph $G$ of order $n\ge f(n_F)$, \\(1) if $n$ is odd, then $$\lambda_2(G)\le\rho^*\left(\frac{n-1}{2},F\right)$$ with equality if and only if $G\in \mathcal{I}\big(G^*(\frac{n-1}{2},F),G^*(\frac{n-1}{2},F)\big)$; and\\ (2) if $n$ is even, and $F$ does not contain cut edges, then the graph $G^\dag$ with the maximum second largest eigenvalue satisfies $$\lambda_2(G^\dag)=\rho^*\left(\frac{n}{2},F\right)-o(1)$$ and $G^\dag\in \mathcal{E}\big(H_1,H_2\big)$, where $H_1$ and $H_2$ are $F$-saturated graphs on $\frac{n}{2}$ vertices. In particular, other than a complete graph $K_{r+1}$, when $F$ is a book graph $B_{k+1}$ or an odd cycle $C_{2k+1}$, we are able to determine the maximum second largest eigenvalue for $F$-free connected graphs of given order, and completely characterize the extremal graphs.

Let $G$ be a locally finite multigraph that is bipartite and "unbalanced," meaning that it has a nontrivial bipartition $(A,B)$ with $\mathrm{deg}(x) > \mathrm{deg}(y)$ for all $x \in A$ and $y \in B$. We explore matchings in such graphs through the lens of descriptive set theory. In particular, we show that when $G$ is Borel and $\mu$ is a Borel probability measure on its vertex set, there is a Borel matching in $G$ that covers $\mu$-almost every vertex in $A$. This was previously known only under the assumption that $\mu$ is $G$-invariant, which we eliminate using a novel probabilistic approach. We also describe various extra conditions that imply the existence of a Borel matching covering every vertex in $A$. Along the way, we confirm a conjecture of the first and third named authors concerning the existence of Borel independent complete sections in Borel graphs of finite asymptotic separation index. In addition to their intrinsic interest, our results have applications to various other topics, such as edge-colorings, balanced orientations, and equidecomposition theory for group actions. For example, we show that the measurable edge-chromatic number of every Borel multigraph with finite maximum degree $\Delta$ is at most $\lfloor\frac{3\Delta}{2}\rfloor$, matching Shannon's optimal bound for finite multigraphs. Another example is that paradoxical Borel group actions with finite asymptotic separation index admit paradoxical decompositions with Borel pieces. This refines a result of Marks and Unger.

Let $q$ be a prime power and write $\Omega$ for the bigraded algebra of regular differential forms over $\mathbb{F}_q^n$. The general linear group $GL_n(\mathbb{F}_q)$ acts on $\Omega$; write $SI \subseteq \Omega$ for the ideal generated by $GL_n(\mathbb{F}_q)$-invariants with vanishing constant term. The {\em $GL_n(\mathbb{F}_q)$-superspace coinvariant ring} is the quotient $SR := \Omega/SI$. We calculate the bigraded Hilbert series of $SR$ and give an operator-theoretic characterization of the inverse system $SI^\perp$. Our results extend to subgroups $G$ of $GL_n(\mathbb{F}_q)$ which contain $SL_n(\mathbb{F}_q)$.

Let $G$ be a graph with $p(G)$ pendant vertices and $q(G)$ quasi-pendant vertices. Denote by $m_{L(G)}(\lambda)$ the multiplicity of $\lambda$ as a Laplacian eigenvalue of $G$. A graph $G$ is called reduced, if $p(G)=q(G)$. It is known that deleting a pendant path $P_3$ from a graph $G$ cannot change $m_{L(G)}(1)$. By the reduction operation for a graph (defined by Tian and Wong, 2026), we could turn to the reduced graphs with each quasi-pendant vertex of degree 2 to investigate $m_{L(G)}(1)$. Then let $T$ be a reduced tree on $n(\geq 7)$ vertices with each quasi-pendant vertex of degree 2 and without pendant path $P_3$. We first prove that \begin{equation*} m_{L(T)}(1)\leq \frac{n-5}{6} \end{equation*} and the extremal trees attaining the upper bound are determined completely. In addition, let $G$ be an arbitrary connected reduced graph with order $n\geq 6$ and size $m$. Denote by $c=m-n+1$ the first Betti number of $G$, then we obtain \begin{equation*} m_{L(G)}(1)\leq c+\frac{n-2}{4}, \end{equation*} and the extremal graphs attaining the upper bound are characterized completely.

An $n$-partite graph is a graph such that every vertex has a color in $\{1,\ldots,n\}$ and every two vertices of the same color are not adjacent. We study the model comparisons of the theories of $n$-partite graph and $K_{\overline{m}}$-free $n$-partite graph, where $K_{\overline{m}}$ is a complete graph of a given size. The model companion of the theory of $n$-partite graph is simple and has IP. The model companion of the theory of $K_{\overline{m}}$-free $n$-partite graph has $\operatorname{TP}_2$, $\operatorname{SOP}_3$ and $\operatorname{NSOP}_4$ if $n > 2$. Forking independence coincides with dividing independence in this theory.

We show that the category of orthomodular posets is a full coreflective subcategory of the category of strong orthoposets, those orthoposets in which any two orthogonal elements have a join. This coreflection is obtained by building from a strong orthoposet $P$, an orthomodular poset with the same underlying set and same orthocomplementation as $P$, but with modified order. This coreflector restricts to a functor from the category of ortholattices to the category of orthomodular posets, and this functor is right adjoint.

Barcode scans, clear phone calls, reliable data storage, satellite communication, and large-scale quantum computation are all made possible by error correction. We present a handbook version of The Error Correction Zoo, a curated reference of methods for protecting classical or quantum information from errors during storage and transmission. The handbook includes descriptions of these error-correcting codes and a classification according to the symbols they use. It also catalogues relations among codes and related objects such as sphere packings, lattices, designs, groups, and classical and quantum phases of matter. The collection is intended both as a rigorous reference and as a practical aid for tracing the web of code relationships and uncovering new connections.

For many trees $T$, the Ramsey number of $T$, denoted by ${\mathcal R}(T)$, is determined by the sizes of the partition classes in its unique bipartition. In 1976, Burr proved that when $T$ has partition classes of size $t_1$ and $t_2$ with $t_1 \le t_2$, the Ramsey number is at least $\max(2t_2-1,2t_1+t_2-1)$, and conjectured that this is tight. While counterexamples have been found for some pairs $(t_1, t_2)$, a main focus of research on this problem has been determining ratios $t_2/t_1$ or bounds on the maximum degree of $T$ for which Burr's bound is either exactly or asymptotically tight. We essentially resolve these questions for lopsided trees. Specifically, we show that (a) there are counterexamples whenever $t_2 \ge 2t_1$, with the order of magnitude of the difference between the largest Ramsey numbers and Burr's bound being $\max \left( t_1^2/t_2, \sqrt{t_1} \right)$, and (b) for $t_2 \ge 500 t_1$, Burr's bound is tight when $\Delta(T) \le t_2 - t_1$, but is off by at least $C \log t_2$ (even when $t_2 \ge 2 t_1$) when $\Delta(T) \gtrsim t_2 - t_1$. In particular, this shows that Burr's bound need not hold for $t$-vertex trees $T$ with $\Delta(T) \approx t/3$.

This paper studies a lattice structure for almost perfect matchings on certain planar, bipartite (plabic) graphs embedded in a disk. Postnikov's boundary measurement map, and subsequent related work, yielded that plabic graphs parameterize positroid cells within the totally nonnegative Grassmannian with the map itself given in terms of almost perfect matchings with fixed boundary condition. For finite planar bipartite graphs, Propp introduced a distributive lattice structure on their set of perfect matchings. Subsequently Muller–Speyer, provided this distributive lattice structure on the aforementioned almost perfect matchings with fixed boundary condition. Their work also identified the extremal matchings of this lattice for boundary conditions that coincide with face labels of the plabic graph given by the positroid structure. We extend this by giving an explicit construction of extremal matchings in terms of height functions and show that all possible boundary conditions of an almost perfect matching can be obtained within this construction.

The Terwilliger algebras of association schemes over an arbitrary field $\mathbb{F}$ were called the Terwilliger $\mathbb{F}$-algebras of association schemes in [10]. In [7], He and Jiang studied the Terwilliger $\mathbb{F}$-algebras of factorial association schemes. In this paper, we continue studying the Terwilliger $\mathbb{F}$-algebras of factorial association schemes. We get all block idempotents of the Terwilliger $\mathbb{F}$-algebras of factorial association schemes. We get the $\mathbb{F}$-dimensions, the centers, the Jacobson radicals of the block algebras of the Terwilliger $\mathbb{F}$-algebras of factorial association schemes.

A central open question in extremal design theory is Nash-Williams' Conjecture from 1970 that every triangle-divisible graph on $n$ vertices (for $n$ large enough) with minimum degree at least $0.75 n$ has a triangle decomposition. In this paper, we prove this conjecture in full. In 2016, Barber, Kühn, Lo, and Osthus proved that if the fractional relaxation of Nash-Williams' Conjecture holds for minimum degree $cn$ for some constant $c\ge 0.75$, then Nash-Williams' Conjecture holds for any constant $c' > c$. The previously best-known bound on the fractional relaxation was due to Delcourt and Postle from 2021 with $c= \frac{7+\sqrt{21}}{14} \approx 0.82733$. This bound on the fractional relaxation has grown in importance over the years as it has been directly tied to bounds for a number of other problems in extremal design theory. This paper consists of three parts. In Part I, our first main result is a proof of the Fractional Nash-Williams' Conjecture: if $G$ is a graph on $n$ vertices with minimum degree at least $\frac{3n}{4}$, then $G$ has a fractional triangle decomposition. In Part II, our second main result is a Fractional Stability Theorem for Nash-Williams' Conjecture: if a graph $G$ on $n$ vertices has minimum degree close to $\frac{3n}{4}$ but no fractional $K_3$-decomposition, then $G$ is close (in edit distance) to the join of two $\frac{n}{4}$-regular graphs each on $\frac{n}{2}$ vertices. We use this to prove that if a triangle-divisible graph $G$ on $n$ vertices has minimum degree close to $\frac{3n}{4}$ but no $K_3$-decomposition, then $G$ is close (in edit distance) to the join of two $\frac{n}{4}$-regular graphs each on $\frac{n}{2}$ vertices. In Part III, our final main result is a proof of Nash-Williams' Conjecture in full.

In this paper, we prove that for every $k$ and every graph $H$ with $m$ edges and no isolated vertices, the Ramsey number $R(C_k,H)$ is at most $(k-1)m+1\le km$. This settles a problem of Erdős, Faudree, Rousseau and Schelp, which is listed as problem 34 in the graph theory collection.

For $\boldsymbol{\alpha} = (\alpha_1, \dots, \alpha_k) \in \mathbb{F}_2^k$, an $\boldsymbol{\alpha}$-town is a set family in which every $i$-wise intersection has parity $\alpha_i$. Denote by $f_{\boldsymbol{\alpha}}(n)$ the maximum size of an $\boldsymbol{\alpha}$-town on $[n]$. The classical oddtown and eventown problems study the cases $\boldsymbol{\alpha} = (1, 0)$ and $(0, 0)$, respectively. We determine the sharp asymptotics of $f_{\boldsymbol{\alpha}}(n)$ for all $\boldsymbol{\alpha}$, answering questions of Johnston–O'Neill and Wei–Zhang–Ge. We also study a symmetric variant $g_{\boldsymbol{\alpha}}(n)$, in which $i$-wise intersection sizes $|F_1 \cap \dots \cap F_i|$ are replaced by $i$-wise intersection-union sizes $|F_1 \cap \dots \cap F_i| + |F_1 \cup \dots \cup F_i|$.

Spherical $G$-parking functions are a distinguished subset of standard monomials, arising from the skeleton ideals of the $G$-parking function ideal. Explicit enumeration formulas for spherical $G$-parking functions are known only for a few classes of graphs. In this paper, we consider a family of graphs $G_{\ell}$ ($1\leq \ell \leq n-2$), obtained from the complete graph $K_{n+1}$ by deleting the $\ell$ edges joining vertex $1$ to the vertices in $F_{\ell}= \{n-\ell+1, \ldots, n\}$. The uprooted spanning trees of $G_{\ell}-\{0\}$ correspond to the set $\mathcal{U}_n^{1\not\sim F_{\ell}}$ of uprooted trees with vertex set $[n]$ in which vertex $1$ is not adjacent to any vertex in $F_{\ell}$, and we establish that $|\mathcal{U}_n^{1\not\sim F_{\ell}}| = (n-1)^{n-\ell-2}(n-2)^{\ell}(n-\ell-1)$. We derive this formula combinatorially and independently recover it as an application of the matrix tree theorem, obtaining some combinatorial identities as consequences. Finally, we determine the number of spherical $G_{\ell}$-parking functions as $|\mathrm{sPF}(G_{\ell})| = (n-1)^{n-3}(n-\ell-1)^2$.

A signed graph is a graph with signatures ($+1$ or $-1$) on its edges. A cycle is called positive if the product of its edge signatures is positive, and a signed graph is called balanced if each cycle in it is positive. The frustration index is the minimum number of edges whose deletion makes the signed graph balanced, which is considered to be a measurement of the imbalance of the signed graph. In this paper, we compare the frustration index of the all-negative signature with the maximum frustration index of all possible signatures on the unsigned graph. We classify some families of graphs into three scenarios: the all-negative signature does not maximise the frustration index, the all-negative signature maximises the frustration index non-uniquely, and the all-negative signature maximises the frustration index uniquely. For all three scenarios, we can exhibit chordal and non-chordal graphs alike. The classes we consider include apex trees, fan graphs, wheel graphs, and complete split graphs. Moreover, for the families of fan graphs and wheel graphs, we fully characterise and count the signatures maximising the frustration index. Throughout our study, we exhibit different classes of signed graphs for which the frustration index equals the number of edge-disjoint negative triangles. Moreover, as part of our study, we are able to refute three conjectures of Zaslavsky on the frustration index.

A long-standing conjecture in the emerging discrete Bakry-Émery theory asserts that bounded-degree graphs satisfying $\mathrm{CD}(0,\infty)$ have polynomial growth. In the present paper, we prove this conjecture for all edge-regular graphs, and even obtain a volume doubling estimate with a constant that depends only on the degree. This is made possible thanks to the discovery of a surprising self-improvement phenomenon, which seems of independent interest: any edge-regular graph satisfying $\mathrm{CD}(\kappa,\infty)$ for some $\kappa\in\mathbb R$ must in fact satisfy $\mathrm{CD}(\kappa,n)$ for some explicit, universal and optimal dimension parameter $n$.

Our paper shows that if a matroid contains the Fano plane or its dual as a minor, then its toric ideal does not have any quadratic Gröbner basis. More than 25 years ago, Hibi, Herzog, and Sturmfels established a direct connection between the existence of quadratic Gröbner bases and regular unimodular flag triangulations. Our paper solves a famous question posed by Herzog and Hibi on a polyhedral reformulation for the existence of quadratic Gröbner bases: we show that the base polytopes of the Fano plane and its dual do not have regular unimodular flag triangulations which implies the main result on Gröbner bases. Our proof relies on several novel tools: a lemma that connects the $1$-skeleton of a lattice polytope to the lattice points in its dilations, an encoding with Boolean formulas and SAT solvers, and symmetry-breaking arguments.

Let $a_2''(n)$ and $b_2''(n)$ be the signed partition functions introduced by Andrews and El Bachraoui for interval-restricted partitions whose parts greater than $1$ are controlled by the smallest even part and by the number of ones. We prove two conjectures for these functions. The first gives the generating function for $a_2''(n)$ as an elementary rational term plus a false theta series with periodic signs; the second asserts that the companion coefficients $b_2''(n)$ take only the values $-1,0,1,2$. The central structural result introduces an auxiliary variable $z$ recording the number of non-compulsory parts greater than $1$. We obtain closed forms for the two resulting generating functions and prove the master identity $(1+q^2)\mathcal B(z,q)-(1+q)\mathcal A(z,q)=-q^4/(1-q^3)$ using both analytic and combinatorial techniques. At $z=-1$, this identity, together with a Rogers–Fine evaluation, gives the false theta formula for $a_2''(n)$ and an explicit generating function for $b_2''(n)$. The latter formula implies the asserted coefficient range and leads to an exact coefficient description of $b_2''(n)$. We also include a direct Heine–Rogers–Fine proof of the false theta formula, ordinary and fixed-refinement consequences of the master identity, and the resulting quantum modular interpretation.

Let $G$ be a finite abelian group of order $n$, and let $\mathsf M(G,m)$ denote the set of zero-sum sequences over $G$ of length $m$. We introduce the zero-sum polytope $\mathcal P_G$, a rational polytope of dimension $n-1$, whose lattice points encode zero-sum sequences: \[ |\mathsf M(G,m)|=|m\mathcal P_G\cap \mathbb Z^n|. \] This naturally realizes the enumeration of zero-sum sequences as a problem in rational Ehrhart theory, which leads to a combinatorial reciprocity theorem identifying the negative evaluations of the corresponding counting quasipolynomial with zero-sum sequences of full support. Our main results establish a face-stratified rigidity for zero-sum polytopes: whenever two such polytopes have equal total lattice point counts at specific dilations, the dimension-wise open-face strata are equinumerous. Moreover, we study the natural $\operatorname{Aut}(G)$-action on $\mathcal P_G$, derive equivariant generating functions and reciprocity formulas, and obtain cyclic sieving phenomena for natural cyclic actions.

The Graph Burning Problem (GBP) is a NP-Hard combinatorial optimization problem that models the propagation of influence or contagion in a network. The propagation is represented through the metaphor of a fire spreading through the vertices of a graph. A burning process takes place in a series of discrete time-steps. At each time step, the burning process is characterized by a propagation (where burned nodes spread the fire to their neighbors), and an ignition (where one additional unburned node is chosen to become burned). The minimum number of steps required to burn all vertices of a graph defines its burning number. Literature provides integer linear programming formulations to solve the problem, but with no surprise, such approaches struggle to converge as the graph size increases. Therefore, reducing the search space explored by these formulations becomes a key point to improve performances. In this work, we study the similarities of the Graph Burning Problem with the well-known Dominating Set Problem. We propose a new formulation based on this study and apply dominance rules and symmetry-breaking techniques to reduce the search space and consequently speed up resolution time. We also introduce a perturbation of the proposed objective function, as well as a pruning rule for the perturbed model in order to further accelerate its resolution.

We present eighty-nine optimal degree-diameter Cayley graphs that are non-isomorphic to previously known optimal Cayley graphs with the same degree and diameter. For each graph, we provide comprehensive data on its parameters and structural invariants. Comparison with known constructions reported by Marston Conder on the Combinatorics Wiki and in related works reveals significant differences in girth, algebraic connectivity, domination number, automorphism group structure, cycle distributions, distance-related properties, and other graph invariants. These findings show that optimal degree-diameter Cayley graphs are frequently non-unique. The new examples enlarge the catalogue of known optimal graphs and provide benchmarks for investigations in degree-diameter problems, algebraic graph theory, and extremal graph theory. Since some of the graphs were identified through randomized search techniques, the collection is not exhaustive.

For every prime power $r\ge 2$, we show that any Sperner family $\mathcal F\subseteq 2^{[n]}$ with $|\mathcal F|\ge (r-1)n+1$ contains $r$ pairwise disjoint nonempty subfamilies whose unions are all equal and whose intersections are all equal. For $r=2$, this confirms a conjecture of Hegedüs, with the sharp threshold $n+1$. In this purely combinatorial problem, our proof combines a multilinear polynomial method, a continuity argument, and the topological Tverberg theorem.