Papers

Let $G$ be a finite group acting properly by lattice automorphisms on a complete simplicial fan $\Sigma$. An open question due to Stanley asked whether the (ungraded) representation carried by the cohomology $H^*(X_{\Sigma})$ of the associated toric variety $X_{\Sigma}$ is isomorphic to a permutation representation of $G$. We prove that Stanley's question has an affirmative answer for all smooth projective toric varieties without the properness assumption on the action. The proof is inspired by toric mirror symmetry.

Let $G$ be a $3$-partite graph with $k$ vertices in each part such that the bipartite graph induced by any two parts contains no cycle of length four. Fischer and Matoušek [J. Combin. Theory Ser. A, 2001] asked for the maximum number of triangles in such a graph. They obtained the lower bound $(1-o(1))k^{3/2}$ and the upper bound $k^{7/4}+O(k^{3/2})$. Coulter, Matthews and Timmons [J. Combin. Theory Ser. B, 2018] later constructed such graphs using planar polynomials over finite fields and improved the lower bound to $(1-o(1))k^{5/3}$. In this note, we use a new triple of planar polynomials and further improve the lower bound to $(1-o(1))k^{17/10}$.

Let $C \le \mathbb{F}_q^n$ be a uniformly random $\mathbb{F}_q$-linear code of rate $1 - h_q(\rho) - \varepsilon$, and let $L^*(C,\rho)$ be the least $L$ such that every Hamming ball of relative radius $\rho$ contains at most $L$ codewords of $C$. That $L^* = \Theta_{q,\rho}(1/\varepsilon)$ has been known since work of Guruswami, Håstad and Kopparty and of Guruswami and Narayanan. Guruswami, Li, Mosheiff, Resch, Silas and Wootters proved that the constant in front of $1/\varepsilon$ is at least $h_q(\rho)$ for all $q$, along with an upper bound special to $q = 2$ which narrowed $L^*$ to within three consecutive integers in that case. But for $q \ge 3$ no upper bound with the correct constant was known. We determine $L^*$ for every prime power $q$. Let $\zeta := h_q(\rho)/\varepsilon$. For every sufficiently small $\varepsilon$, with probability $1-o(1)$ over the choice of $C$, $$L^*(C,\rho) = \lceil \zeta \rceil,$$ unless the fractional part of $\zeta$ is at most $q^{-\Omega_{q,\rho}(\zeta)}$, in which case $L^*(C,\rho)$ is $\lfloor \zeta \rfloor$ or $\lfloor \zeta \rfloor + 1$. By the threshold characterization of random linear codes due to Mosheiff, Resch, Ron-Zewi, Silas and Wootters, both bounds reduce to a two-sided estimate of a single quantity $V(q,L,\rho)$, where $1-V(q,L,\rho)$ is the threshold rate for $(\rho,L)$-list-decodability. We prove for all large $L$: $$h_q(\rho)(1 + 1/L) - q^{-\Omega_{q,\rho}(L)} \le V(q,L,\rho) \le h_q(\rho)(1 + 1/L).$$ The upper bound rests on a new entropy inequality for sparse random vectors under pairwise non-proportional linear constraints, proved with the Erdős-Rado sunflower lemma. The lower bound is an exact analysis of the distribution introduced by Guruswami, Li, Mosheiff, Resch, Silas and Wootters.

We determine the closure of the connected domination roots. The main tool is a substitution formula for the lexicographic product with a complete graph, $D_c(G[K_n],x)=D_c(G,(x+1)^n-1)$, proved in Theorem~\ref{thm:cd}. Combined with two explicit families of seed roots—the real roots of the cycles $C_n$ and the real roots of the joins $C_m\vee C_n$ lying in $(-1,0)$—this formula gives the two main results: the closure of the real connected domination roots is $(-\infty,0]$, and the closure of all connected domination roots is the whole complex plane. These are the connected domination analogues of the root-density theorems of Brown and Tufts and of Brown and Beaton for the ordinary domination polynomial.

In a paper in Acta Mathematica Hungarica the author proved that the Morley tetrahedron of an isosceles tetrahedron, obtained by trisecting the six dihedral angles, is again isosceles, and proposed two converse conjectures. We show that both are false. There is a nonisosceles tetrahedron $T_1$ and an isosceles, nonregular tetrahedron $T_2$ whose Morley tetrahedra are regular, and there are nonisosceles tetrahedra, even a two-parameter family of tetrahedra without any symmetry, whose Morley tetrahedra are isosceles. In $T_1$ and in $T_2$ there is a pair of opposite edges such that the other four edges are equal, and we conjecture that a regular Morley tetrahedron always forces this. We prove the conjecture for every tetrahedron with a nontrivial symmetry, and we show that, up to similarity, the regular tetrahedron, $T_1$ and $T_2$ are the only tetrahedra with this edge pattern and a regular Morley tetrahedron. The tetrahedron $T_2$ has $AB=CD=1$ and $AC=AD=BC=BD=\sqrt{(21+4\sqrt6)/45}$, while $T_1$ is given by a root of a sextic with Galois group $S_6$ and cannot be expressed by radicals. We also prove that a tetrahedron with a regular Morley tetrahedron is regular if it is orthocentric, if it is isodynamic, if its three sums of opposite edges are equal, if it has three equal edges at a vertex, if it has an equilateral face, or if none of its dihedral angles is larger than $95^\circ$; the tetrahedron $T_2$ has two dihedral angles of about $98.7^\circ$. For isosceles Morley tetrahedra we conjecture that $(AB^2-CD^2)(AC^2-BD^2)(AD^2-BC^2)\ge0$ and that $AB=CD$ forces a second pair of equal opposite edges. We also conjecture that a tetrahedron with $AC=BD$ whose Morley tetrahedron satisfies $A'B'=B'C'=C'D'=D'A'$ has a nontrivial symmetry. Some proofs are computer assisted; they use exact rational arithmetic or interval arithmetic with outward rounding.

In this note, we show that the Hilbert scheme $\text{Hilb}_{P_{d,4}(t)}(E_6/P_6)$ associated with the Hilbert polynomial $P_{d,4}(t)$ is disconnected by determining that it has exactly two connected components. This result adapts Seong's methods, successfully extending the disconnectedness of Grassmannians to the exceptional type $E_6$.

We develop a spread-based approach to finding induced cycles and apply it to two problems. First, we resolve the odd-hole gadget conjecture of Bradač, Draganić and Sudakov by constructing an $e^{O(k)}$-edge graph whose every $k$-edge-colouring contains a monochromatic induced odd cycle of length $O(\log k)$. As a consequence, for every $k\ge2$ and every sufficiently large odd $n$, $$ \widehat R_{\mathrm{ind}}(C_n;k)=e^{\Theta(k)}n. $$ The proof uses spread probability weights together with hypergraph containers. Second, we prove that for every sufficiently large fixed $d$, with high probability the largest hole in the random $d$-regular graph $G_{n,d}$ has order $\Theta(n\log d/d)$, resolving a problem of Frieze. Although the two proofs use different mechanisms, both begin with a well-distributed auxiliary object and use it to control the extra edges that could destroy inducedness.

Let \(\beta(G)\) denote the minimum size of a feedback edge set of a digraph \(G\), and let \(\gamma(G)\) denote the number of unordered pairs of nonadjacent vertices. Motivated by the Chudnovsky–Seymour–Sullivan conjecture for \(3\)-free digraphs, we study the corresponding feedback-edge problem for bipartite digraphs. In the bipartite setting, \(\gamma(G)\) is taken to count only nonadjacent pairs with ends in distinct partite sets. We prove that every \(4\)-free bipartite digraph \(G\) satisfies \(\beta(G)\le \gamma(G)/2\). We also determine the exact Turán number of \(2k\)-free strong bipartite digraphs with partite sets \(X\) and \(Y\): if \(|X|,|Y|\ge k+1\), then the maximum number of edges is $$(|X|-(k-1))(|Y|-(k-1))+2k-2.$$ Finally, for the extremal case \(k=2\), we analyze the structure of \(4\)-free strong bipartite Turán digraphs and prove the sharper bound \(\beta(G)\le \gamma(G)/3\) for all such digraphs. This constant is attained by a natural balanced three-block construction.

Let $S_k(G)$ denote the sum of the $k$ largest Laplacian eigenvalues of a connected graph $G$ of order $n$ and size $m$. Write $\mathrm{PA}_{n,\omega}$ for the graph obtained from an $\omega$-vertex clique by attaching $n-\omega$ pendant vertices to one of its vertices, and set \[ M_{n,k}:=\binom{k+1}{2}+n-k-1, \] the number of edges of $\mathrm{PA}_{n,k+1}$. For $n/2<k\le n-2$, we prove the sharp bound \[ S_k(G)\le \frac{2k}{n}m+ \frac{2(n-k)}{n}M_{n,k}-(n-k-1), \] with equality attained by $\mathrm{PA}_{n,k+1}$. This bound is complementary to Brouwer's inequality and is strictly stronger when $m<M_{n,k}$. Combining our bound with Brouwer's inequality, we resolve and strengthen a conjecture of Vinagre, Del-Vecchio, Justo, and Trevisan: for every $n$, the pineapple $\mathrm{PA}_{n,1+\lfloor2n/3\rfloor}$ maximizes the Laplacian energy among all connected graphs of order $n$; moreover, for $n>4$, it is the unique maximizer.

For a graph \(H\), the \(k\)-color Ramsey number \(r(H;k)\) is the least integer \(N\) such that every \(k\)-edge-coloring of \(K_N\) contains a monochromatic copy of \(H\). A \(k\)-list assignment has \(|L(e)|=k\) for every edge. The list Ramsey number \(r_\ell(H;k)\) is the least integer \(N\) for which there exists a \(k\)-list assignment on \(E(K_N)\) such that every coloring from the lists contains a monochromatic copy of \(H\). Let \(K_{1,n}\) be a star, \(S(n,m)\) the double star obtained by joining the centers of \(K_{1,n}\) and \(K_{1,m}\), and \(S_n^m\) the graph obtained from \(K_{1,n}\) by subdividing \(m\) edges once. Alon et al.\ conjectured that \(r_\ell(K_{1,n};k)=r(K_{1,n};k)\) for all \(k,n\ge1\). In this paper, we confirm their conjecture for all \(k\ge1\) and \(n\ge3\) by a unified direct proof. For even \(k\ge4\) and under explicit parameter conditions, we prove that \(r(S(n,m);k)=kn+m+2\) for even \(n\) and \(r(S_n^m;k)=k(n-1)+m+2\) for odd \(n\). For two colors, we establish a general list Ramsey lower bound and determine the common values of Ramsey and list Ramsey numbers for double and subdivided stars in explicit parameter ranges. These results close several gaps in the known bounds.

A degree-$d$ polynomial source is the output of a polynomial map of degree at most $d$ over $\mathbb{F}_2$ on arbitrarily many uniform random bits. Khodabandeh and Shinkar (FOCS '26) proved that $\mathrm{Ber}(1/3)^{\otimes N}$ has statistical distance $1-o(1)$ from every constant-degree polynomial source and conjectured exponentially small overlap. Independently of Khodabandeh and Shinkar, Byramji, Kane, Morris, and Ostuni (RANDOM '26) asked for an explicit target distribution at distance $1-\exp(-N^{\Omega_d(1)})$. We resolve both questions. For every fixed $d\geq1$, every degree-$d$ polynomial source has overlap at most $\exp(-c_dN)$ with $\mathrm{Ber}(1/3)^{\otimes N}$, where $c_d>0$ is independent of the seed length. For quadratics, $c_2=2^{-26}$ suffices. We amplify Khodabandeh and Shinkar's uniform separation of acceptance probabilities from non-dyadic parameters (numbers not of the form $a/2^b$ for integers $a$ and $b\geq0$). The result extends to other non-dyadic Bernoulli parameters and to coordinates that are Boolean functions of boundedly many bounded-degree polynomials. We also give a uniform deterministic hierarchy between adjacent degrees. Appending the outputs of disjoint AND gates on $d+1$ inputs to uniform seed bits yields flat degree-$(d+1)$ target distributions of entropy $k$ with overlap $\exp(-\Omega_d(\min\{k,N-k\}))$ against every degree-$d$ source, for $\min\{k,N-k\}\geq2(d+1)$. This entropy dependence is optimal up to constants in the exponent among flat target distributions for fixed $d$. The construction has locality $d+1$ and uses $O(N)$ field operations to sample. At $k=\lfloor N/2\rfloor$, it handles $d\leq(1-\varepsilon)\log_2N/3$ with overlap $\exp(-N^{\varepsilon-o(1)})$ for fixed $0<\varepsilon<1$. The proof combines monotonicity of Gowers uniformity norms, pairwise independence of points in a random affine cube, and relative entropy.

We study the generic nullity of generalized commutator operators \[L_{\mathbf{A}}(X)=s_{k+1}(A_1,\cdots,A_k,X)\] on matrix algebras, where $s_{k+1}$ denotes the standard polynomial. Dixon and Pressman conjectured an explicit formula for the generic nullity of $L_{\mathbf{A}}$, and Brassil and Reichstein proved the conjecture when $k$ is even. In this paper, we settle the remaining case where $k$ is odd. Our proof first treats the boundary cases $k=2n-3$ in dimensions $n$ and $n+1$ using degree decompositions and graph-theoretic interpretations of alternating trace forms, and then establishes a dimension-extension argument from $n$ to $n+2$. Consequently, together with the result of Brassil and Reichstein, we obtain a complete proof of the Dixon-Pressman generic nullity conjecture over any field of characteristic zero.

For a graph $G$ of order $n$, its positive, negative and non-positive inertia is the number of positive, negative and non-positive eigenvalues of its adjacency matrix $A(G)$, respectively. Mohar asked whether every graph with $k$ non-positive eigenvalues has order $O(k^2)$ as $k\to \infty$. Using NEPS, we construct a sequence of non-singular graphs with negative inertia $k$ and order $\Omega(k^{\frac{7}{3}})$ as $k\to \infty$, thus resolving Mohar's problem. Our result also strongly refutes a recent conjecture of Akbari, Elphick, Kumar, Pragada, and Tang involving positive and negative inertia.

We study several $k$-fold generalizations of the Lovász theta function associated with the maximum $k$-colorable induced subgraph problem. The first is the Narasimhan–Manber parameter $\vartheta_k$. We prove that, for graphs whose adjacency matrix belongs to a homogeneous partially coherent algebra, this parameter is recovered by the theta number of the Cartesian product with the complete graph on $k$ vertices. This class includes distance-regular and $1$-walk-regular graphs, and thus our result generalizes a theorem by Sinjorgo and Sotirov (2022) for graphs that are vertex- and edge-transitive. We introduce a new parameter $\varphi_k$ obtained from orthonormal representations of graphs and show the inequality $\varphi_k \leq \vartheta_k$. For both parameters, we study the smallest $k$ for which the parameter is equal to the number of vertices; these saturation parameters yield lower bounds on the chromatic number. We determine which vertex-weighted versions of these parameters are gauges, and discuss a natural definition for the $k$-fold theta body of a graph. We conclude with open questions comparing $\vartheta_k$, $\varphi_k$, $\vartheta(G\square K_k)$, and related convexifications.

For a finite set $A$ of positive integers, let $H(A)$ be its set of subset sums, and let $g_3(n)$ be the least $N$ for which some $n$-element set $A\subseteq\{1,\ldots,N\}$ has $H(A)$ free of nonconstant three-term arithmetic progressions. Erdős and Sárkőzy asked whether $g_3(n)\gg 3^n$. We prove \[ \liminf_{n\to\infty}\frac{g_3(n)}{3^n}=0. \] More precisely, for every $\epsilon>0$ there is an integer $d\ge2$ such that $g_3(d\ell)\le \epsilon 3^{d\ell}$ for all sufficiently large $\ell$. The proof uses Korsky's characterization of the problem in terms of ternary coefficient sums and a consequence of an OpenAI construction that provides positive integer coefficients whose linear form is injective on large integer boxes. A base-three expansion then gives the result.

If $\boldsymbol{w} = (w_1,w_2,\dots)$ is a sequence in $\mathbb{N}=\{1,2,\dots\}$, the partial Bell polynomials based on $\boldsymbol{w}$ are $B_{n,k}$ for $k \in \mathbb{N}$ and $n\in\{k,k+1,\dots\}$. Let $F(z) = \sum_{n=1}^{\infty} (w_n/n!)z^n$ be the exponential generating function for $\boldsymbol{w}$, and assume the radius of convegence is positive $R>0$. Then $F(z)^k = \sum_{n=k}^{\infty} (k!/n!) z^n B_{n,k}$ for $|z|<R$. Alternatively, defining $Q_{k,n} = (k!/n!)B_{n,k}$, we have $Q_{1,n} = w_n/n!$, and $Q_{k+1,n}=\sum_{m=1}^{n-k} Q_{1,m} Q_{k,n-m}$ for $k\geq 1$. Let us say that the Cramér-type large deviation property holds if $$ \lim_{\substack{n \to \infty\\ k/n \to \kappa}} \frac{1}{n}\, \ln\left(Q_{k,n}\right)\, =\, \mathcal{G}(\kappa)\, ,$$ for every $\kappa \in (0,1)$, where $\mathcal{G}(\kappa)=\inf_{r \in (0,R)} (\kappa \ln(F(r))-\ln(r))$. The (Hardy-Ramanujan) Erdös induction argument suggests this should generally be true as long as two technical conditions are true: one an initial step, and the other a condition for small densities $\kappa$.

We study the Lonely Runner Conjecture (LRC), conceived by Jörg M. Wills in the 1960's: Given positive integers $n_1, n_2, \dots, n_k$, there exists a positive real number $t$ such that for all $1 \le j \le k$ the distance of $t \,n_j$ to the nearest integer is at least $\frac{ 1 }{ k+1 }$. We prove that for any counterexample or tight instance $\mathbf{n}$ of LRC, $\mathbf{m} \cdot \mathbf{n} = 0$ for some $\mathbf{m} \in \mathbb{Z}^k$ with $0 < \| \mathbf{m} \|_1 \le \min(2k+3, \ \frac{ k+1 }{ k-1 } \mathrm{flt}(k))$ where $\mathrm{flt}(k)$ denotes Khinchin's (1948) flatness constant limiting the lattice width of a $k$-dimensional convex body without interior integer points. In other words, potential counterexamples to LRC lie on a finite set of hyperplanes in the parameter space. Our proofs use Fourier analysis and a geometric reformulation of LRC, and our results generalize to the situation of shifted lonely runners of varying measures of loneliness. Our results imply and generalize a theorem of Czerwiński (2012) that when we choose $\mathbf{n}$ at random then, with probability tending to 1, the measure of loneliness $\frac{1}{ k+1 }$ can be replaced by $\frac 1 2 - \epsilon$.

Schwartz-MacPherson classes of degeneracy loci of symmetric and skew-symmetric maps. Together with the known formulas for ordinary linear maps, this completes the stable SSM theory for the three classical types. We interpret these formulas probabilistically, in terms of the endpoint random partition of a stochastic rational six-vertex model: the skew classes are probabilities of Maya-dimer events, and the symmetric classes are Chebyshev moments of the random 2-core.

We study random planar quadrangulations in which each block – i.e., each simple (without multiple edges) component – is assigned a weight $u$. We derive an explicit expression for the distance-dependent two-point function, defined as the generating function of such block-weighted maps with two marked edges at a fixed graph distance. Using contour integral representations in two variables combined with a delicate saddle-point analysis, we compute the associated distance profile in the scaling limit of large quadrangulations. We recover the known phase transition at $u = 9/5$, characterized by distinct scaling exponents and different scaling functions, below, at, and above criticality. We also discuss the block distance profile, where the two marked edges are conditioned to lie within the same block.

We study Gallai decomposition for groups equipped with two-sided invariant partial orders. The key algebraic step extends to arbitrary binary relations compatible with the group operation: if all left and right translations preserve a binary relation $\rho$, then every least strong module $S_{\rho}(\e,g)$, $g\ne\e$, is a subgroup. For a partial order this subgroup is convex. Thus the robust modules through the identity of an ordered group form a canonical chain of convex subgroups, with each canonical factor $H/H^-$ prime, totally ordered, or equality-ordered. We characterize exactly the subgroups that are modules, show that they form a complete sublattice of the subgroup lattice, establish overlap and inheritance results for arbitrary subgroups, and prove compatibility with quotients by normal strong subgroups. For $N$-free ordered groups the prime factors disappear. Using the robust-module decomposition of cographs, we characterize all two-sided invariant $N$-free partial orders by reduced admissible two-coloured subgroup chains, with totally ordered and equality-ordered canonical factors; the order is determined by the first nontrivial factor of each element. We also determine how the canonical decomposition restricts to arbitrary subgroups, characterize finite width and prove width divisibility for subgroups, and show that every reduced two-coloured chain is realized by an $N$-free ordered abelian group whose canonical factors are isomorphic to $\mathbb Z$.

We show that, for some absolute constants $c_1,c_2>0$, every $(n,d,\lambda)$-graph with $\lambda\le c_1 d$ contains an induced cycle of length at least $c_2n\log(d/\lambda)/d$. This is best possible up to the values of $c_1,c_2$. Our techniques include a multi-scale algorithmic analysis, an adapted depth-first exploration procedure, a link to percolation theory, and estimates for random row-and-column extraction in symmetric matrices.

Let $f(w,k)$ be the maximum size of a $w$-uniform family containing no sunflower with $k$ petals. We introduce a recursive construction for sunflower-free families and use it to obtain a general lower bound on the exponential growth rate of $f(w,k)$. We also prove a general upper bound for $3$-uniform families with at least four petals. Our results give $39\le f(3,4)\le49$, $f(3,5)\le146$, $153\le f(3,6)\le255$, $259\le f(3,7)\le474$, and $54\le f(4,3)\le83$. In addition, we prove that the maximum size of an intersecting $4$-uniform family containing no sunflower with three petals is $27$. The upper bounds $49$ and $83$ are computer-assisted. The finite lower bounds come from explicit constructions.

We study bounded ratios and optimal bounding constants among the normalized coefficients of ternary Lorentzian polynomials. For every fixed $M$-convex support and in arbitrary degree, we give an explicit presentation of the bounded-ratio cone in terms of quadratic Hessian slices. We then express the optimal bounding constants through a variational formula combining local support functions with linear compatibility constraints between slices. For full support, we determine all compatibility relations in arbitrary degree; in degree three, this yields explicit optimal constants for every two-generator section. Finally, we compare the resulting Lorentzian bounds with those for volume polynomials and rank-three matroid basis profiles.

Let $P_d(132,213)$ be the convex hull of the permutations in $S_d$ that avoid $132$ and $213$, and let $H_d(t)$ be its Ehrhart $h^*$-polynomial, defined by $\sum_{m\ge0}|mP_d(132,213)\cap\mathbb{Z}^d|t^m=\frac{H_d(t)}{(1-t)^d}$. We give an explicit lattice equivalence between this polytope, a path-Laplacian deficit polytope, and twice a translated chain-poset permutahedron. The resulting face description and triangulation give a self-contained proof of Davis and Sagan's cubicality and volume conjecture, with normalized volume $2^{d-1}d^{d-3}$ for $d\ge2$. For $d\ge3$, the coefficient of $t^{d-1}$ in $H_d(t)$ counts strong tournament score sequences. The refined Eulerian expansion of $H_d(t)$ has a negative coordinate already at $d=7$. We construct a compatible cone of adjacent signed differences that accommodates this obstruction. Its largest uniform repair parameter is $2-\sqrt{3}$, and membership reduces to explicit geometric-tail inequalities. An exact dataset and reconstruction algorithm verify these inequalities through $d=1000$. A marked-component identity, discrete smoothing, and Darroch's mode theorem (Ann. Math. Statist. 35 (1964), 1317–1321, Theorem 4) prove them for every $d\ge2^{72}$. Thus $H_d(t)$ has only negative real zeros for $3\le d\le1000$ and for $d\ge2^{72}$, while $H_1(t)=H_2(t)=1$. Uniform real-rootedness in the intervening range remains open.

Let $T_n$ be the infinite $n$-regular tree, $n\ge3$. We prove that every finite connected vertex set $\Omega\subset T_n$ satisfies the sharp inequality \[ \sigma_1(\Omega)\le \frac{n}{(n-1)|\Omega|+1}. \] Equality holds if and only if $\Omega$ is a ball. Since \[ |\delta\Omega|=(n-2)|\Omega|+2, \] the result is equivalently a sharp upper bound at fixed external boundary cardinality, and hence a discrete Weinstock inequality on $T_n$.

The Bott-Duffin theorem shows that any arbitrary positive real rational function of finite order, representing a passive network, can be represented by a finite number of positive value resistors, capacitors and inductors. It is shown here that if the positivity restriction is lifted, any rational function or polynomial of any order may be represented by a network of (possibly negative) inductors, capacitors and resistors. When there are no repeated roots this is a sum of simple sub-circuits each of which contains no more than four components. When there are repeated roots a more complex circuit is required, here featuring iterated Wheatstone bridges of special form.

Let $m\geq0$ and put $n=m+3$. We introduce a normalized positive Laurent class of decorated frieze patterns and prove a decorated analogue of the Conway-Coxeter classification theorem. Namely, such friezes of width $m$ are in canonical bijection with weighted triangulations of a convex $n$-gon, whose boundary edges are labeled by independent variables $y_1,\ldots,y_n$ and whose diagonals are labeled by $x_1,\ldots,x_m$. While a weighted Conway-Coxeter propagation algorithm constructs the frieze from a weighted triangulation, a main new ingredient is an explicit nonrecursive Laurent formula expressing each quiddity entry directly from the weighted triangles incident to the corresponding vertex. Conversely, specialization to $1$, Laurent positivity, and a decorated cutting-and-gluing procedure recover the weighted triangulation from the frieze.

In this paper, we investigate the difference between ordinary and symbolic powers of edge ideals on the Castelnuovo-Mumford regularity and the depth. As a main theorem, we prove that the regularities of ordinary and symbolic powers coincide for edge ideals of simplicial graphs, as a partial result of Minh's conjecture. For the depth, we first show that, for each $k\geq2$, the inequality $\operatorname{depth} S/I^{(k)}\geq\operatorname{depth} S/I^{k}$ does not hold for squarefree monomial ideals in general. On the other hand, we prove that it holds for edge ideals when $k=2,3$. We also study the symbolic-ordinary discrepancy module $I(G)^{(k)}/I(G)^{k}$ of the edge ideal $I(G)$ of a graph $G$ and give a graph-theoretic formula for its Krull dimension in terms of induced odd cycles of $G$, thereby answering a question and a problem posed by Ha and Minh.

We summarize the classical passage from the chromatic specialization of Negami's polynomial to the Hilbert series of a broken-circuit Stanley–Reisner ring, and record formulas for several graph families with their connectivity hypotheses made explicit. Our main construction in this revision is the associated graded algebra of the squarefree edge algebra under the graphic-matroid rank filtration, for an arbitrary finite loopless graph. Its basis is indexed by all edge subsets, and its multiplication is governed by disjointness and rank additivity. Its bigraded Hilbert polynomial recovers the full three-variable Negami polynomial. We prove that its socle has a basis indexed by edge subsets of full matroid rank; for a connected graph, these are precisely the connected spanning subgraphs. A vertex-degree refinement counts connected regular spanning subgraphs. This finite-dimensional algebra is distinct from a broken-circuit Stanley–Reisner ring. We distinguish these elementary structural results from the unresolved question of whether abstract algebra isomorphism can distinguish graphs with the same Negami polynomial; no priority claim is made for the construction or its Hilbert identity.

For an abelian group $\Gamma$ of order $n$, a graph $G$ of order $n$ is $\Gamma$-distance magic if it admits a bijection $V(G)\to\Gamma$ whose open-neighborhood sums are constant, and group distance magic if this holds for every such $\Gamma$. More generally, for any finite abelian $\Gamma$, a $\Gamma$-magic map is a map $f\colon V(G)\to\Gamma$ with constant open-neighborhood sums; we call $f$ generating if its labels generate $\Gamma$, and affinely generating if its pairwise differences do. Let $G$ be regular, let $\Gamma\cong\mathbb{Z}/d_1\oplus\cdots\oplus\mathbb{Z}/d_r$ with $d_1\mid\cdots\mid d_r$, put $B_\Gamma=\bigoplus_{i<r}\mathbb{Z}/d_i$, and let $\overline{A}_m$ be the adjacency operator induced on $(\mathbb{Z}/m)^{V(G)}$ modulo constants. We prove that $G$ admits a generating $\Gamma$-magic map iff $B_\Gamma\hookrightarrow\ker\overline{A}_{d_r}$, and an affinely generating one iff $\Gamma\hookrightarrow\ker\overline{A}_{d_r}$; when $|V(G)|=|\Gamma|$, it is $\Gamma$-distance magic iff the latter embedding has vertex-separating image. If the reduced adjacency operator is nonsingular over $\mathbb{Q}$, these become subgroup conditions in the reduced adjacency Smith group. Cichacz and Froncek conjectured that every distance magic graph is group distance magic. Using the affine criterion we construct a $6$-regular distance magic graph of order $27$ admitting no affinely generating $(\mathbb{Z}/3)^3$-magic map, so the conjecture fails even for affine generation. We propose the generating group distance magic conjecture, and prove it for regular distance magic graphs whenever $\Gamma$ is $2$-generated, hence for cube-free order. Further applications concern Cayley graphs on elementary abelian $p$-groups, Hamming relation graphs, strongly regular graphs, and symmetric designs.