Papers

In this article, we construct noncommutative cluster varieties, $\mathcal{A}_{R,S}$, for each reduced root system $R$ and marked surface $S$ simultaneously generalizing the cluster varieties of Fock-Goncharov, Li, Goncharov-Shen, Berenstein-Retakh, Goncharov-Kontsevich, and our previously introduced polygonal cluster algebras. Additionally, we define a large class of algebraic groups, we call Jordan split groups. Given a reduced root system $R$ and a family of Jordan algebras, the Lie algebra for $G$ is constructed by unifying the Tits-Kantor-Koecher construction for a single Jordan algebra with the construction of a split Lie algebra. The notion of Jordan split groups is closely related to a grading of its Lie algebra by the root system $R$. We show that these gradings are usually induced by a choice of standard parabolic subalgebra $\mathfrak{p}_\Theta$ and we classify $R$-graded pairs $(G,\Theta)$ via a condition depending only on the subset $\Theta\subset \Delta$ of the set of simple roots. Next, we define Jordan algebra points of $\mathcal{A}_{R,S}$ which parameterize $G$-local systems on $S$ with boundary decoration related to cosets $G/U_\Theta$ when $G$ is Jordan split of type $R$. When $S$ is a disk, points of $\mathcal{A}_{R,S}$ parameterize configurations of decorated flags. We use this to give noncommutative cluster structures on the double $R$-Bruhat cells of $G$, generalizing the cluster algebras of Berenstein-Fomin-Zelevinsky. When each Jordan algebra is formally real, we say that $G$ has a positive structure with respect to $\Theta$. This defines a positive semigroup in $G$. For real algebraic groups, the pairs $(G,\Theta)$ which have positive structures are exactly those which admit a positive structure as defined by Guichard-Wienhard and we give algebraic proofs of many of the properties of positive configurations of flags and of positive representations.

In this paper, we establish a structural compatibility between the Kronecker product of bipartite graphs that admit polarity and their polarity quotient, and provide a sharp upper bound on the diameter of these graphs. For certain factor graphs, the diameter of the Kronecker product meets the upper bound on diameter, among them the generalized polygons. Generalized polygons with their polarity quotients have been notably used in the past to construct very large graphs. We apply the structural theorems in the paper to generalized polygons $\mathbb{G}_n(q,q)$ used as factor graphs, and build three new families of graphs of large order covering an infinite but sparse set of degrees, one of diameter $2$, one of diameter $3$ and one of diameter $5$. These asymptotically approach a theoretical upper bound on graph size as orders $q$ and $r$ of the generalized polygon factors increase. As an example, we develop one such family, derived from generalized quadrangles, and construct new diameter-$3$ graphs of low degree that are larger than any previously known at their degrees.

A long-standing question posed by Fukuda (1995) and Ziegler (2000) inquires about the asymptotic behavior of $g(n)$, the maximum number of facets that an $n$-dimensional $0/1$-polytope can have. A remarkable result by Bárány and Pór (2001) via probabilistic methods established that $g(n)$ is at least superexponential in $n$. In this paper, we propose a drastic change of perspective, which leads us to show that for each $n\geq 10$ there exists a $0/1$-polytope having at least $(n-\lceil 2\log_2 (n)\rceil - 1)!$ facets. This provides a significant improvement over the currently known lower bounds for $g(n)$. Furthermore, when combined with known upper bounds, our construction establishes the asymptotic behavior of $\log g(n)$ up to an error of $O((\log n)^2)$. The methods employed throughout this paper are elementary and fully deterministic. The underlying ideas in our proof stem from the combinatorics of hypersimplices and permutohedra.

We study the maximum average quantum transition probability between distinct vertices of a triangle-free graph, equivalently the minimum average return probability, at times inversely proportional to the order. For every compact interval of positive scaled times below $\tau_{\mathrm c}\approx3.79099$, the balanced complete bipartite graph is the unique maximizer for all sufficiently large orders. The threshold is sharp: it solves $\tau_{\mathrm c}=4\sin(\tau_{\mathrm c}/2)$, and the balanced complete bipartite graph is eventually suboptimal at every larger scaled time. The corresponding graphon functional combines the edge density with alternating even cycle densities. We prove that the balanced complete bipartite graphon is its unique maximizer through the critical time. On a nonempty interval immediately afterward, the unique maximizer is a balanced bipartite graphon with constant cross-edge weight below one. After normalization by the squared time, we obtain uniform square-root $L^1$ stability through the critical endpoint and at zero time. We also solve the bipartite problem at every positive time and establish sharp cut distance stability. The proofs combine spectral interpolation, an exactly certified six-vertex moment inequality, vertex-measure variation, and a minimum-degree argument. A uniform finite approximation bound and vertex cloning connect the graphon and exact finite results.

Given a group $G$ and three subsets $S_\ell, S_r, S_m \subset G$, we consider di-Cayley graphs $DX(G;S_\ell,S_r,S_m)$ and di-Cayley sum graphs $DX^+(G;S_\ell,S_r,S_m)$, directed generalizations of the bi-Cayley (sum) graphs $BX(G;S_\ell,S_r,S_m)$ and $BX^+(G;S_\ell,S_r,S_m)$. We refer to these four kinds of graphs collectively as $X^*(G;S_\ell,S_r,S_m)$. First, we give the basic properties of these graphs and compute their adjacency matrices. Then, we obtain the eigenvalues of $X^*(G;S_\ell,S_r,S_m)$ in terms of the spectra of the associated Cayley graphs $X(G,S)$ with $S\in \{S_\ell,S_r,S_m\}$ in two ways, using adjacency matrices and using irreducible characters of $G$.

In this paper, we compute the (shifted) Hankel determinants of Catalan-like sequences, which arise naturally from the weighted enumerations of nonintersecting Motzkin meanders. Among these determinant evaluations, one and a half are newly discovered, featuring generic shifted Hankel determinants; two were formulated earlier by Cigler and Krattenthaler in an equivalent combinatorial form; and the rest were conjectured by Cigler. As an application, we further confirm a conjectural binomial determinant identity proposed by Cigler and Krattenthaler.

We study versions of results in combinatorial geometry related to families of convex sets in $\mathbb{R}^d$ whose intersection contains a $k$-dimensional affine space. We prove generalizations of the colorful Radon theorem, the fractional Helly theorem, the colorful Helly theorem, and the selection-structure Helly theorem. When $k=0$, our arguments give new proofs of the corresponding versions for points.

In this paper, we resolve a 30-year-old conjecture of Spielman and Teng concerning the performance of the spectral partitioning method on graphs embeddable on an orientable surface of genus $g\ge 1$. In particular, for such a graph $G$ with $n$ vertices and maximum degree $\Delta$, we show that the second-smallest eigenvalue of its Laplacian matrix satisfies $\lambda_2(L_G)\lesssim\Delta\frac g n$. We also obtain an improved eigenvalue bound for $K_h$-minor-free graphs of $\lambda_2(L_G)\lesssim\Delta\frac{h^2(\log h)^2}n$. In fact, our results directly prove much stronger results for reweighted eigenvalues, including higher reweighted eigenvalues. As a consequence, we obtain bounds not just on Laplacian eigenvalues, but also on normalized Laplacian eigenvalues and Steklov eigenvalues. Our results for genus-$g$ graphs are optimal for all of these kinds of eigenvalues, while our results for $K_h$-minor-free graphs are optimal up to $\log(h)$ factors. Our techniques for genus-$g$ graphs bootstrap bounded-degree bounds of normalized eigenvalues for entire classes to bounds for reweighted eigenvalues for the same classes without the bounded-degree limitation, while our techniques for $K_h$-minor-free graphs generalize an argument of Korhonen and Lokshtanov, making use of the Lovász local lemma.

We present a simple combinatorial method for computing the socle of representations of $\mathrm{GL}_n(F)$, where $F$ is a local non-Archimedean field, parabolically induced from two ladder representations satisfying a permissibility condition. We adapt recent approaches to such problems using the classical Robinson–Schensted–Knuth correspondence (RSK) and introduce a new variant of RSK that we call mixed-RSK. Some combinatorial properties of this map are investigated and then used to resolve a conjecture of Erez Lapid.

A functional digraph is a directed graph where each vertex has an out-degree of at most 1. We study the number of successor-closed subsets of a functional digraph, that is, subsets from which no edge leaves. We show that functional digraphs have a simple recursive formula for their corresponding generating function. Using this formula, we determine, among all functional digraphs with a fixed number of vertices and edges, the one that maximizes and the one that minimizes the number of successor-closed subsets of every size simultaneously. Somewhat unexpectedly, this extremal result yields a quantitative strengthening of the set-pairs inequality of Bollobas: rather than merely guaranteeing that some pair of a large enough family must violate the hypothesis of the theorem, we show that a uniformly random subset of the family witnesses a violation with high probability, quantitatively in terms of how far the family size exceeds the classical threshold. We further show that the same approach applies to the skew variant of Bollobas's inequality due to Hegedus and Frankl, yielding an analogous probabilistic strengthening.

We say that an antichain in a lattice $L$ is Boolean if it generates a Boolean sublattice in $L$ in a particularly nice way. These purely combinatorial objects play a role in the representation theory of the incidence algebra of the lattice $L$, as indicated by recent results of Rognerud, Y{\i}ld{\i}r{\i}m and of the last author with Klász, Kleinau and Marczinzik. Given these motivations, it is natural to ask: Can we count and construct Boolean antichains in certain lattices? In this paper we give a construction of Boolean antichains in Boolean lattices, partition lattices and Tamari lattices. Furthermore, we share the surprisingly elegant formulas we found for the number of these Boolean antichains.

Motivated by the recent introduction of chromatic persistent homology, we generalize the Euclidean minimum spanning tree (EMST) for $n$ points in $\mathbb{R}^2$ to the lunar EMST for the case in which the points come in $s+1$ colors. Calling the intersection of $s+1$ disks of radius $r$ centered at points with pairwise different colors a lune, the generalized EMST reflects the history of the union of lunes as $r$ goes from $0$ to $\infty$, and its cost is twice the difference between the radii when the arcs and nodes of the tree are formed. If the points are chosen uniformly at random in $[0,1]^2$ and colored randomly, the expected cost converges to some constant (that depends on $s$) times $\sqrt{n}$, as $n$ goes to infinity. The main contribution of this paper is a proof that this constant exists, however similar to the case of the classic EMST, its precise value remains elusive.

In this paper, we study the structures of single connected $k$-leaky $(r+1)$-completed Hurwitz numbers. We first prove that, for fixed genus, after extracting an explicit product of Pochhammer type factors, the stable connected leaky Hurwitz numbers for partition $\mu$ are polynomials in the quotients $[\mu_i]$ modulo $k+r$. We then give a closed formula for the generating function of single connected leaky Hurwitz numbers, which can be used to study the genus dependence of leaky Hurwitz numbers.

We show that the strength as a spherical design of the spherical embedding of a $P$- and $Q$-polynomial association scheme with at least three classes with respect to a $Q$-polynomial idempotent is at most five, provided that the multiplicity is at least three. We also identify the examples that attain this upper bound on the strength. Our result improves on Suda's earlier upper bound of eight [J. Combin. Des. 19 (2011)], and is considered dual to the results of Lewis [Discrete Math. 223 (2000)] and Miklavič [Electron. J. Combin. 32 (2025)] concerning the girth of a $Q$-polynomial distance-regular graph with diameter and valency both at least three. To establish our upper bound, we introduce and discuss a polynomial method that works by constructing an appropriate polynomial that vanishes at every point of the spherical embedding.

For a graph $H$, the chromatic threshold $\delta_\chi(H)$ is the infimum of $c>0$ such that the chromatic number of every $n$-vertex $H$-free graph with minimum degree at least $cn$ is bounded by a constant depending only on $H$ and $c$. Allen, Böttcher, Griffiths, Kohayakawa, and Morris proved that if $\chi(H)=r\geq 3$, then $\delta_{\chi}(H)\in\{\frac{r-3}{r-2}, \frac{2r-5}{2r-3}, \frac{r-2}{r-1}\}$. Liu, Shangguan, Skokan, and Xu introduced the bounded-VC chromatic threshold $\text{VC}(H)$ by restricting the host graphs to have bounded VC-dimension. We determine this parameter for graph $H$ with $\chi(H)\ge 3$. More precisely, let $\mathcal{M}(H)$ be the decomposition family of an $r$-chromatic graph $H$, then \[ \text{VC}(H)= \begin{cases} \dfrac{r-3}{r-2},&\text{if $\mathcal{M}(H)$ contains a forest},\\[4pt] \dfrac{r-2}{r-1},&\text{otherwise}. \end{cases} \]

We classify the facial exposedness of completely positive cones over symmetric cones in terms of the rank of the associated Euclidean Jordan algebras. The completely positive cones are facially exposed when the rank is at most $2$, but are not facially exposed when the rank is at least $5$. Facial exposedness is not completely determined by the rank when the rank is $3$ or $4$, but we provide a characterization in each case. The resulting classification of facial exposedness in fact agrees with the corresponding classification of spectrahedrality for completely positive cones.

Let $E_n$ denote the number of alternating permutations of $\{1,\dots,n\}$, equivalently characterized by $\sum_{n\ge0}E_nz^n/n!=\sec z+\tan z$. For every $q\ge1$, the sequence $(E_n\bmod q)_{n\ge0}$ is eventually periodic; let $d(q)$ and $s(q)$ denote its minimal eventual period and preperiod. For every odd prime $p$, Knuth and Buckholtz proved $d(p)=\operatorname{lcm}(p-1,4)$ together with \[ d(p^r)\mid p^{r-1}d(p), \qquad s(p^r)\le r, \] and Ramassamy conjectured that both bounds are attained for every $r\ge1$. In this paper, we introduce an algebraic frequency expansion for the Euler zig-zag numbers using Hurwitz series over the coefficient ring $S_r=(\mathbb Z/p^r\mathbb Z)[x]/(x^2+1)$. More precisely, the corresponding Hurwitz series is represented as a finite combination of formal exponential modes, in a manner reminiscent of Fourier analysis. Using this expansion, we prove \[ d(p^r)=p^{r-1}d(p) \qquad \text{for every odd prime $p$ and every $r\ge1$}, \] thereby establishing Ramassamy's period conjecture. We also disprove the preperiod conjecture by proving \[ s(5^5)=4<5. \] Finally, we prove that $5^5$ is the smallest odd prime power for which $s(p^r)\ne r$, and based on our findings we further conjecture \[ s(p^r)\ge r-2 \] for every odd prime $p$ and every $r\ge2$.

We investigate a combinatorial notion of orientability for abstract GKM graphs and its connections to graph cohomology in the sense of Guillemin–Zara. In particular, we prove that orientability of the GKM graph is equivalent to Poincaré duality of the rational (non-equivariant) graph cohomology algebra. As an application, we prove that orientable GKM graphs remain connected after removing any single vertex.

Let $W$ be an affine Coxeter group of type $\widetilde{A}_n$, that is, the affine symmetric group $\widetilde{S}_N$ with $N=n+1$, let $T$ be its set of reflections, and let $\mathrm{Red}_T(w)$ be the set of reduced reflection factorizations of an element $w\in W$. The braid group acts on $\mathrm{Red}_T(w)$ by the Hurwitz action. For finite Coxeter groups it is known exactly when this action is transitive, namely precisely for the parabolic quasi-Coxeter elements. We address this problem for the affine type $\widetilde{A}_n$ by determining all orbits of $\mathrm{Red}_T(w)$.

Let $G$ be a finite simple graph and let $S\subseteq V(G)$. We prove that the minimum number of vertices meeting every cycle that intersects $S$ is at most the maximum number of vertices of $S$ covered by a collection of vertex-disjoint cycles. This answers a question posed by Bowler, Ghorbani, Gut, Jacobs, and Reich [SIAM Journal on Discrete Mathematics 40 (2026), 988–999]. An incidence-based reduction to their bidirected packing–covering theorem preserves the packing value and projects transversals without increasing their cardinality.

Letzter, Methuku and Sudakov proved that sufficiently regular sublinear expanders contain nearly spanning cycles and paths. Montgomery subsequently conjectured that every regular sublinear expander of sufficiently large degree is hamiltonian. Recently, Chen, Liu, Wei and Yang disproved this conjecture by constructing $d$-regular sublinear expanders of order $n$ with $d=(1/2+o(1))\log^2 n$ and small circumference. They further posed the problem of determining whether, for every fixed $\varepsilon>0$, there exists a constant $C=C(\varepsilon)$ such that every sufficiently large $n$-vertex $d$-regular $(\varepsilon,d)$-expander with $d\ge C\log^2 n$ is hamiltonian. We answer this problem negatively by constructing, for every sufficiently small fixed $\varepsilon>0$, infinitely many nonhamiltonian $d$-regular $(\varepsilon,d)$-expanders with degrees far above the $\log^2 n$ scale.

Let ${\cal G}=\{G_1,\ldots,G_k\}$ be a collection of (not necessarily distinct) bipartite graphs on the same vertex bipartition $(X,Y)$, where $|X|=a$, $|Y|=b$ and $2\le k\le a\le b$. A rainbow matching of ${\cal G}$ is a set of pairwise disjoint edges that can be chosen from distinct members of ${\cal G}$. Denote by $q(G)$ the signless Laplacian spectral radius of a graph $G$. In this paper, we prove that if $q(G_i)\ge b+k-1$ for each $i\in\{1,2,\ldots,k\}$, then ${\cal G}$ admits a rainbow matching of size $k$ unless $G_1=\cdots=G_k\cong K_{k-1,b}\cup\overline{K_{a-k+1}}$, and show that the threshold is sharp and attained by the exceptional collection. The condition is also extended to larger collections for a prescribed level $t$. In addition, we obtain a lower bound for the rainbow matching number in terms of the ordered signless Laplacian spectral radii of the members, and provide a stability version of the extremal characterization. In the proofs, we use the shifting technique and a quotient matrix arising from an equitable partition of a signless Laplacian matrix.

We show that for every sufficiently large genus $g$, there exists a closed hyperbolic surface $S_g$ with systole $\mathrm{sys}(S_g)\geq \log g-12\log\log g$. In particular, $$\liminf_{g\to \infty}\frac{\max\{\mathrm{sys}(S):S\in \mathcal{M}_g\}}{\log g}\geq 1,$$ improving the previously known bound $2/9$. This note is a continuation of our previous work on the diameter of finite covers arXiv:2608.12887, using the same framework of constant-twist pants decomposition to study systoles. The proof was developed by GPT-5.6 Sol through an extended discussion with the author.

Let $\mathcal{Z}_w^{(\mathbb{F}_q)}$ be the $\mathbb{F}_q$-linear subspace of $\mathbb{F}_q(\!(\theta^{-1})\!)$ spanned by Thakur's multiple zeta values $\zeta_A(\mathfrak{s})$ of weight $w$. We prove that $\sum_{w=1}^{\infty} \left(\dim_{\mathbb{F}_q} \mathcal{Z}_w^{(\mathbb{F}_q)}\right) x^w = \frac{x(1-x^q)(1-2x+x^q)}{(1-2x+x^{q+1})^2}$. Moreover, we construct an explicit $\mathbb{F}_q$-basis of $\mathcal{Z}_w^{(\mathbb{F}_q)}$, and prove that any $\mathbb{F}_q$-linear relation among Carlitz multiple polylogarithm values $\operatorname{Li}_A(\mathfrak{s})$ is an $\mathbb{F}_q$-linear combination of quadruple-carry relations. This result can be regarded as an $\mathbb{F}_q$-analogue of the corresponding $\mathbb{F}_q(\theta)$-theorem proved by Chang–Chen–Mishiba and independently by Im–Kim–Le–Ngo Dac–Pham. Our discovery of the quadruple-carry relations is inspired by the recent work of Im–Kim–Ngo Dac. These relations may be viewed as $\mathbb{F}_q$-analogues of the double-shuffle relations among classical multiple zeta values $\zeta(\mathfrak{s})$.

Classical Turán theory shows that, without additional constraints, the extremal structure of $F$-free graphs is governed by the chromatic number of $F$ and is asymptotically the balanced Turán graph. We investigate how prescribing a large maximum degree changes this picture and leads to an unbalanced Turán problem. Let $F$ be a graph with $\chi(F)=r+1\ge3$, and let $\lceil(r-1)n/r\rceil\le\Delta\le n-1$. We study the maximum number of edges and adjacency spectral radius of an $n$-vertex $F$-free graph with maximum degree exactly $\Delta$. For $F=K_{r+1}$, the unbalanced $r$-partite graph $S_{n,\Delta}^{(r)}=(n-\Delta)K_1\vee T(\Delta,r-1)$ is the unique maximizer of both quantities. Let $a(F)$ be the minimum size of an independent set $I$ such that $\chi(F-I)\le r$. If $a(F)=1$, we prove edge and spectral stability with respect to $S_{n,\Delta}^{(r)}$. If $a(F)>1$, the extremal values are $t(n,r)+o(n^2)$ and $\rho(T(n,r))+o(n)$, respectively, and every extremal graph differs from $T(n,r)$ in $o(n^2)$ edges, uniformly all $\lceil(r-1)n/r\rceil\le\Delta\le n-1$. We further establish a general spectral transfer principle: for finite forbidden families, a decomposition-family bound of order $O(n^{1+s})$ yields a spectral bound with error $O(n^s)$ for $0\le s<1$.

The line graph $L(G)$ of a graph $G$ is defined to be the simple graph whose vertices are the edges of $G$, where two vertices in $L(G)$ are adjacent if and only if the corresponding edges in $G$ are incident with a common vertex, and define $L^2(G)=L(L(G))$. For positive integers $d$ and $k$, the function $\kappa_{L^2}(d,k) = \inf\{\kappa(L^2(G)): \kappa'(G) \ge k \mbox{ and } \delta(G) \ge d\}$ has been investigated. Niepel and Knor proved that $\kappa_{L^2}(d,1)\geq d-1$, for any integer $d \ge 3$. In this research, it is proved that if $d\geq 3$ and $k\geq 1$, then $\kappa_{L^2}(d,k)= \min\{f(d,k), 4d-6\}$, where \begin{equation} f(d,k) = \left\{ \begin{array}{ll} k(d-k), & \mbox{ if $1\leq k\leq \lfloor\frac{d}{2}\rfloor$, } \\ kd-k^2+2k(\lceil \frac{d}{2}\rceil)-(\lceil\frac{d}{2}\rceil)d, & \mbox{ if $\lfloor\frac{d}{2}\rfloor< k <\frac{3d-1}{4}$, } \\ kd-k^2+2k\lfloor \frac{d}{2}\rfloor-2(\lfloor \frac{d}{2}\rfloor)^2, & \mbox{ if $\frac{3d-1}{4}\leq k<d$, }\\ d(\lceil \frac{d}{2}\rceil), & \mbox{ if } d=k. \end{array} \right.\nonumber \end{equation}

We record five improved lower bounds for Zarankiewicz numbers with s = t = 3: z(13,19;3,3) >= 118, z(14,19;3,3) >= 126, z(16,18;3,3) >= 136, z(14,20;3,3) >= 126, z(16,19;3,3) >= 136. The first three are certified by explicit K_{3,3}-free 0/1 matrices; the last two follow from the second and third by monotone padding. Compared with the lower bounds compiled in Figure 2 of arXiv:2605.01120, namely 114, 121, 130, 125 and 132, the improvements are +4, +5, +6, +1 and +4 respectively. The three matrices were produced by a kicked-greedy local search operated autonomously by a discovery system and were verified exactly, by exhaustive inspection of every 3x3 row/column triple, in several mutually independent implementations; they were also re-verified independently by the authors of arXiv:2605.01120 using their own verifier in July 2026. All three witnesses are printed in full in Appendix A and accompany this note as machine-readable ancillary files together with standalone, dependency-free verifiers.

This article studies identifying codes in oriented graphs from a computational complexity perspective. We investigate the $\mathcal{F}$-Id Code problem, where given a simple graph $G$ and a vertex subset $C$, which induces a subgraph in the family $\mathcal{F}$, as inputs and ask whether it is possible to orient $G$ in such a way that $C$ becomes its oriented identifying code. Focusing on the family $\mathcal{F}_d$ of $d$-regular graphs, we establish a complete dichotomy by proving that the problem is polynomial-time solvable for $d\leq1$ and NP-complete for all $d\geq2$.

We investigate random recursive simplicial complexes growing by adding, at each step, a vertex together with a simplex formed by joining the new vertex with a randomly chosen existing simplex. We also add all faces of the new simplex to ensure that the resulting object remains a simplicial complex. If the choice of an existing simplex is uniform among simplices of dimension $<m$, the number $S_d$ of simplices of any admissible dimension $d\leq m$ is an asymptotically self-averaging random variable. This feature allows us to determine the asymptotic growth law of the average of $S_d$ when the number of vertices diverges. We also probe the degree distribution, examine the probabilities of various extreme outcomes, and analyze the characteristics of the first vertex.

We study decoding procedures for a family of MDS array codes previously constructed from the Kronecker product of a superregular matrix and a non-singular matrix over a finite field. By exploiting the particular structure of their parity-check matrices, we develop decoding algorithms for different channel models. For the erasure channel, we provide an algorithm capable of recovering any pattern of up to $n-k$ symbol erasures. For the $q$-ary symmetric channel, we investigate the decoding of one and two symbol errors and give explicit procedures for determining their locations and values. We also consider the particular case in which the superregular matrix is a Vandermonde matrix, showing how its additional algebraic structure can be exploited in the decoding process. Explicit examples over different finite fields are provided to illustrate the proposed procedures.