arXiv subjects

Yinfeng Zhu

Publications and source records attributed to Yinfeng Zhu.

7 recordsLinked to original sources

arXiv2026

Paths maximize the expected range of graph-indexed random walks

We prove that a path maximizes the expected range of a uniformly chosen graph homomorphism into the integers, with one vertex pinned at zero, among all connected bipartite graphs of the same order. This establishes the expectation form of the Benjamini--Häggström--Mossel conjecture. The proof restricts and rescales a homomorphism on each bipartition class, then contracts the edges on which the resulting height function is constant. A quantitative estimate for the rank of these zero edges compensates for a parity term in the expected range of a simple random walk, allowing an induction on the number of vertices. We then prove that the BHM inequality implies the Loebl--Ne\v set\v ril--Reed inequality for uniformly chosen integer 1-Lipschitz functions on arbitrary connected graphs, and hence obtain the LNR conjecture as a corollary of BHM. The proof was obtained through interaction with OpenAI GPT-6 Astra and verified by the author. The main results have also been formalized and checked in Lean~4.

math.CO↗

arXiv2026

Completely Reachable Road Coloring

We characterize the digraphs that admit an edge labeling by letters from a finite alphabet such that the resulting labeled digraph is a completely reachable automaton. This class of digraphs can be recognized in polynomial time. In contrast, we show that, for every fixed alphabet size, the problem of deciding whether a digraph admits an edge labeling with the same property is NP-complete. We also classify the digraphs for which every edge labeling results in a completely reachable automaton.

cs.FL↗

arXiv2026

The Černý Conjecture for One-Cluster Automata via Annular Spectral Descent

We prove the Černý conjecture for synchronizing one-cluster automata. More precisely, let a synchronizing automaton with state set $Q$, $|Q|=n$, have a letter $a$ whose functional digraph has a unique cycle $C$ of length $m$, and let $\ell$ be the least nonnegative integer for which $a^\ell$ maps $Q$ onto $C$. Assume $\ell\ge1$. For every nonempty proper subset $S\subset C$, we prove that there is a word $w$ of length at most $n$ such that $wa^\ell$ maps more than $|S|$ states of $C$ into $S$. This proves the positive-level part of a conjecture of Kisielewicz, Kowalski, and Szykuła concerning relative extending words for one-cluster automata. The resulting reset word has length at most \[(m-1)(n-1)+m\ell\le(n-1)^2. \] For every $n\ge4$, we construct a strongly connected binary example with $m=2$, $\ell=n-2$, and reset threshold $3n-5$, so the parameter-dependent bound $(m-1)(n-1)+m\ell$ is sharp. The upper-bound proof uses finite-dimensional linear algebra; the sharpness lower bounds are combinatorial. The proof was obtained through interaction with OpenAI Codex (GPT-5.6 Sol, ultra mode) and verified by the author.

cs.FL↗

arXiv2024

A quadratic upper bound on the reset thresholds of synchronizing automata containing a transitive permutation group

For any synchronizing $n$-state deterministic automaton, Černý conjectures the existence of a synchronizing word of length at most $(n-1)^2$. We prove that there exists a synchronizing word of length at most $2n^2 - 7n + 7$ for every synchronizing $n$-state deterministic automaton that satisfies the following two properties: 1. The image of the action of each letter contains at least $n-1$ states; 2. The actions of bijective letters generate a transitive permutation group on the state set.

cs.FL↗

arXiv2024

Around Don's conjecture for binary completely reachable automata

A word $w$ is called a reaching word of a subset $S$ of states in a deterministic finite automaton (DFA) if $S$ is the image of $Q$ under the action of $w$. A DFA is called completely reachable if every non-empty subset of the state set has a reaching word. A conjecture states that in every $n$-state completely reachable DFA, for every $k$-element subset of states, there exists a reaching word of length at most $n(n-k)$. We present infinitely many completely reachable DFAs with two letters that violate this conjecture. A subfamily of completely reachable DFAs with two letters, is called standardized DFAs, introduced by Casas and Volkov (2023). We prove that every $k$-element subset of states in an $n$-state standardized DFA has a reaching word of length $\le n(n-k) + n - 1$. Finally, we confirm the conjecture for standardized DFAs with additional properties, thus generalizing a result of Casas and Volkov (2023).

cs.FL↗

arXiv2019

The effect of a graft transformation on distance signless Laplacian spectral radius of the graphs

Suppose that the vertex set of a connected graph $G$ is $V(G)=\{v_1,\cdots,v_n\}$. Then we denote by $Tr_{G}(v_i)$ the sum of distances between $v_i$ and all other vertices of $G$. Let $Tr(G)$ be the $n\times n$ diagonal matrix with its $(i,i)$-entry equal to $Tr_{G}(v_{i})$ and $D(G)$ be the distance matrix of $G$. Then $Q_{D}(G)=Tr(G)+D(G)$ is the distance signless Laplacian matrix of $G$. The largest eigenvalues of $Q_D(G)$ is called distance signless Laplacian spectral radius of $G$. In this paper we give some graft transformations on distance signless Laplacian spectral radius of the graphs and use them to characterize the graphs with the minimum and maximal distance signless Laplacian spectral radius among non-starlike and non-caterpillar trees.

math.CO↗

arXiv2017

Half of an antipodal spherical design

We investigate several antipodal spherical designs on whether we can choose half of the points, one from each antipodal pair, such that they are balanced at the origin. In particular, root systems of type A, D and E, minimal points of Leech lattice and the unique tight 7-design on $S^{22}$ are studied. We also study a half of an antipodal spherical design from the viewpoint of association schemes and spherical designs of harmonic index $T$.

math.CO↗