arXiv subjects

Mohammad F. Marashdeh

Publications and source records attributed to Mohammad F. Marashdeh.

5 recordsLinked to original sources

arXiv2026

Two-basepoint Terwilliger algebras and the quantum symmetry of prime-order circulants

Which vertex-transitive graphs of prime order have quantum symmetry? The question of Banica, Bichon and Chenevier is open in the dense regime of Paley graphs, where coherent-algebra methods give no information. To each such graph we attach a two-basepoint Terwilliger algebra of its cyclotomic scheme and study the module it generates from the basepoints: fullness forces the quantum permutation algebra to be commutative, and the module admits no intermediate state, containing either exactly two point masses or all $p$ of them. One point mass, captured at any depth, therefore suffices, and Chassaniol's orbital criterion is the depth-one case. Three consequences follow. A sharp counting argument replaces the Banica--Bichon--Chenevier threshold $p>6^{φ(k)}$ by the quadratic bound $p>(k-1)(k-2)+2$, where $k$ is the type. Four certificates, each a short list of additions modulo $p$, settle $C_{31}(2,4,8,15)$ and $C_{41}(4,10,16,18)$, the two graphs left open by Chassaniol, and complete the classification for type at most $10$ without machine assistance. An exact computation extends the dichotomy ``quantum symmetry if and only if complete or empty'' to all prime orders $p\le250$, settling the Paley graphs $P_{p}$ with $p\le241$, the first beyond $P_{17}$. What remains is the capture of a single explicit vector: the midpoint $2^{-1}$ of the two basepoints.

math.OA↗

arXiv2026

A square-root law for equitable coloring

An equitable $k$-coloring of a graph partitions its vertex set into $k$ independent sets whose sizes differ by at most one; the least such $k$ is the equitable chromatic number $\chie(G)$. Every known bound on $\chie$ valid for all graphs, beginning with the Hajnal--Szemerédi theorem, is linear in the maximum degree $Δ$, and the star $K_{1,Δ}$, for which $\chie=\ceil{Δ/2}+1$, shows that no general bound below $Δ/2$ exists. We prove that this obstruction is a shortage of vertices rather than an effect of the degree: every graph with $|V(G)|\ge3χ(G)Δ$ satisfies $\chie(G)=O\bigl(χ(G)^{3/2}\sqrt{Δ/\lnΔ}\bigr)$ throughout the range $χ(G)\le(Δ/\lnΔ)^{1/3}$, so that for graphs of large order the degree enters only through $\sqrt{Δ/\lnΔ}$, with the chromatic number governing the rest. For each fixed $\ell$, $\ell$-colorable graphs of sufficiently large order satisfy $\chie\le\bigl(2\sqrt2\,\ell\sqrt{\ell-1}+o(1)\bigr)\sqrt{Δ/\lnΔ}$, while a probabilistic construction supplies graphs of arbitrarily large order, bipartite when $\ell=2$, with $\chie\ge\tfrac13\sqrt{(\ell-1)Δ/\lnΔ}$: the order of growth $Θ\bigl(\sqrt{Δ/\lnΔ}\bigr)$ is exact for every fixed chromatic number, and the extremal constant is determined up to a factor $O(χ(G))$. All upper bounds are constructive, and a prescribed-anchor variant of the construction produces equitable colorings of bipartite graphs with $O(\sqrtΔ)$ colors in optimal linear time.

math.CO↗

arXiv2026

An upper bound for the type of a numerical semigroup, and a reduction of Wilf's conjecture

Let $S$ be a numerical semigroup with multiplicity $\mult$, conductor $\cc$, embedding dimension $\ee$, type $\typ$ and genus $\gnus$, and let $\nn=\cc-\gnus$. Wilf's conjecture asserts that $\ee\,\nn\ge\cc$; the inequality $\gnus\le\typ\,\nn$ of Fröberg, Gottlieb and Häggkvist settles it when $\typ\le\ee-1$. The Apéry set of $S$ with respect to any $s\in S\setminus\{0\}$ carries a partial order whose maximal elements are the pseudo-Frobenius numbers translated by $s$; for $s=\mult$ its minimal elements are the minimal generators other than $\mult$. Comparing the two extremal statistics bounds the type by $\typ\le\ee-1+\Xii(S)\le\ee-1+Θ(S)$, where $Θ(S)$ measures the redundancy of the covering of the gaps of $S$ by the pseudo-Frobenius numbers and $\Xii(S)$ refines it. With an exact decomposition of the Wilf number this yields the genus bound $\gnus\le\ee-1+\typ(\nn-1)$, strictly stronger than $\gnus\le\typ\,\nn$ precisely when $\typ\ge\ee$, and reduces Wilf's conjecture to an inequality free of $\cc$ and $\nn$. We determine the equality case of $\gnus\le\typ\,\nn$, recovering a classification of Singhal; answer a question of Moscariello and Sammartano whenever $\ee\ge\typ+1$; and correct Kaplan's classification of the equality case for $\cc\le2\mult$, from which an infinite family is missing.

math.CO↗

arXiv2026

A descent-set obstruction for the Casas-Alvero conjecture

The Casas--Alvero conjecture asserts that a monic polynomial $f$ of degree $d$ over a field of characteristic zero sharing a non-constant factor with each of $f',\dots,f^{(d-1)}$ is the $d$-th power of a linear polynomial. Two reductions are carried out. Once a root is translated to the origin, the condition at index $i$ is vacuous unless the coefficient $a_{d-i}$ is nonzero, so the Casas--Alvero locus stratifies by the support of the centred normal form; prescribing in addition which root realises each surviving condition makes the system unit lower triangular and eliminates the coefficients of $f$ entirely. When one root suffices, what remains is a single integer, and that integer is MacMahon's determinant: the number of permutations of $\{1,\dots,d\}$ with descent set the support of $f$. Since every subset is a descent set, no counterexample has fewer than three recycled roots; in characteristic $p$ a prime dividing one of these counts but none of the accompanying integers $z_{i}$ is a bad prime, the largest count in degree $d$ being the Euler zigzag number $A_{d}$, so that the irregular prime $691$ is bad for $d=11$. The stratification also yields short proofs of the two characteristic-$p$ propositions on which all known cases rest; supports of two elements are settled separately in every characteristic, whence a counterexample has at least four terms in centred normal form and a further Gröbner-free criterion for bad primes.

math.NT↗

arXiv2026

A phase transition in the directional growth of lamplighter groups

Let $G$ be a finitely generated group with a surjective homomorphism $π\colon G\to\Z$. The directional growth spectrum $I(β)$ is the exponential rate of the number of elements of length at most $n$ lying over $\lfloorβn\rfloor$; its maximum is the growth rate. Wherever $I$ has been computed---free, hyperbolic and relatively hyperbolic groups, free abelian groups---it is strictly concave and real-analytic, as Perron--Frobenius theory dictates. We compute $I$ in closed form for the lamplighter groups $F\wr\Z$, $|F|=r+1$, with the standard generators, and obtain a different picture: $I$ is affine on $[-β^{*},β^{*}]$ and strictly concave beyond it, with a second-order transition at $β^{*}$. Elements conditioned to the affine phase backtrack macroscopically, with lamp density independent of $β$. The series $\sum_x s^{|x|}y^{π(x)}$ is rational, and the transition is an exchange of dominant singularities, the inner one independent of $y$. The peak of $I$ is $\logω$ with $ω^{2}=ω+r$, the central value $\logρ$ with $ρ^{3}=ρ+r$; the base group is therefore co-amenable yet grows strictly slower, by an amount unbounded in $r$. For $r=1$ these constants are the golden ratio and the plastic number.

math.GR↗