arXiv subjects

Tingzeng Wu

Publications and source records attributed to Tingzeng Wu.

At least 19 recordsLinked to original sources

arXiv2026

Solution to an open problem on the computational complexity of immanant

Immanants are a class of generalized matrix functions associated with the irreducible characters of the symmetric group. Bürgisser [SIAM J. Comput., 30 (2000), pp. 1023--1040] proved that the computation of hook immanants and immanants corresponding to rectangular Young diagrams of polynomially growing width is VNP-complete under $p$-projections. And he posed an open problem: whether the family of immanants corresponding to rectangular Young diagrams of width $2$ is VNP-complete under $p$-projections. This paper gives a solution to this problem. We prove that, over any field of characteristic zero, the immanant families associated with rectangular Young diagrams of width $2$ and of length $2$ are both VNP-complete under $p$-projections.

math.CO↗

arXiv2026

On the Greedoid Tutte Polynomial for simple rooted graphs

The Tutte polynomial, through its rank-generating formula, provides a unified framework for describing various combinatorial structures of graphs and matroids, and serves as an important polynomial invariant connecting graph theory, matroid theory, and their related applications. Let $G=(V(G),E(G),r(G))$ be a simple rooted graph, where $V(G)$ is the vertex set, $E(G)$ is the edge set, and $r(G)$ is the root. Let $Γ(G)$ be the greedoid induced by the rooted graph $G$, and let its greedoid Tutte polynomial be denoted by $T(Γ(G);x,y)$. Let $r=r(G)$, $L_r(G)=\{v\in V(G)\setminus\{r\}: rv\in E(G),\ d_G(v)=1\}$, and $\ell_r(G)=|L_r(G)|$. Gordon and McMahon proposed the following conjecture: for rooted graphs, $\operatorname{ord}_{x}T(Γ(G);x,y)=\ell_r(G)$, where $\operatorname{ord}_{x}T(Γ(G);x,y)=\max\{a\in\mathbb Z_{\geq0}:x^a\mid T(Γ(G);x,y)\}$, that is, the highest power of $x$ dividing $T(Γ(G);x,y)$ is equal to the number of leaf vertices adjacent to the root. In this paper, we proved that this conjecture holds for all simple rooted graphs.

math.CO↗

arXiv2026

An improved range for the maximum critically $t$-intersecting hypergraphs

Let $k>t\ge 1$ be integers and set $d=k-t$. A $k$-uniform hypergraph $\mathcal F$ is called $t$-intersecting if any two edges intersect in at least $t$ vertices, and is called $t$-critical if its minimum $t$-transversal has size $k$. Frankl proved that, for $k\ge d^4$,$|\mathcal F|\le \binom{k+d}{d},$ with equality only for the complete $k$-graph on $k+d$ vertices, and conjectured that the same conclusion should hold when $k>c d^2$ for some constant $c$. In this paper we confirm this conjecture for $c=30$. The proof relies on Frankl's fixed-edge decomposition and Füredi's pseudo-sunflower method.

math.CO↗

arXiv2026

Non-uniform pairwise cross $t$-intersecting families

Let $ n\geq t\geq 1$ and $ \mathcal{A}_1, \mathcal{A}_2, \ldots, \mathcal{A}_m \subseteq 2^{[n]}$ be non-empty families. We say that they are pairwise cross $t$-intersecting if $|A_i\cap A_j|\geq t$ holds for any $A_i\in \mathcal{A}_i$ and $A_j\in \mathcal{A}_j$ with $i\neq j$. In the case where $m=2$ and $\mathcal{A}_1=\mathcal{A}_2$, determining the maximum size $M(n,t)$ of a non-uniform $t$-intersecting family of sets over $[n]$ was solved by Katona (1964), and enhanced by Frankl (2017), and recently by Li and Wu (2024). In this paper, we establish the following upper bound: if $ \mathcal{A}_1, \mathcal{A}_2, \ldots, \mathcal{A}_m \subseteq 2^{[n]}$ are non-empty pairwise cross $t$-intersecting families, then $$ \sum_{i=1}^m |\mathcal{A}_i| \leq \max \left\{ \sum_{k=t} ^{n}\binom{n}{k} + m - 1, \, m M(n, t) \right\}. $$ Furthermore, we provide a complete characterization of the extremal families that achieve the bound. Our result not only generalizes an old result of Katona (1964) for a single family, but also extends a theorem of Frankl and Wong (2021) for two families. Moreover, our result could be viewed as a non-uniform version of a recent theorem of Li and Zhang (2025). The key in our proof is to utilize the generating set method and the pushing-pulling method together.

math.CO↗

arXiv2026

Thresholds for the Frankl-Wang $3/7$ conjecture on maximum-degree ratios

Let $\mathcal{F}\subset\binom{[n]}{k}$ be an intersecting family, $Δ(\mathcal{F})=\max_{x\in[n]}|\{F\in\mathcal{F}:x\in F\}|$, and $\varrho(\mathcal{F})=Δ(\mathcal{F})/|\mathcal{F}|$. Frankl and Wang conjectured that if $n>100k$ and $|\mathcal{F}|>\binom{n-3}{k-3}$, then $\varrho(\mathcal{F})\ge 3/7$; the constant $3/7$ is sharp because of the Fano-plane construction. In this note we obtain three results. First, we show that no linear threshold $n>Ck$ can be sufficient: using a truncated Fano-plane construction we exhibit, for every constant $C$ and all large $k$, an intersecting family with $n>Ck$, $|\mathcal{F}|>\binom{n-3}{k-3}$, yet $\varrho(\mathcal{F})<3/7$. In particular, the original condition $n>100k$ does not guarantee the conclusion. Second, for $k=3$ we prove that $\varrho(\mathcal{F})\ge 3/7$ holds for every nonempty intersecting $3$-uniform family; the proof is nontrivial and does not rely on any assumption on $n$ or $|\mathcal{F}|$. Third, using the classical pseudo-sunflower bound $|\mathcal{F}|\le t^k$ (for families containing no pseudo-sunflower of size $t+1$), we obtain a completely explicit polynomial threshold for all $k\ge4$: if $n>(k-3)(7k^4+k)+3$ and $|\mathcal{F}|>\binom{n-3}{k-3}$, then $\varrho(\mathcal{F})\ge 3/7$. In particular, the simplified bound $n>7k^5$ is sufficient for every $k\ge4$.

math.CO↗

arXiv2026

Immanantal polynomials of the linear combination matrices of graphs

In this paper, we focus on the study of immanantal polynomials for linear combination matrices composed of the degree matrix and adjacency matrix of a graph. First, applying the concept of vertex orientation for general graphs, we provide a combinatorial interpretation of the coefficients of the immanantal polynomials for the linear combination matrices of graphs, and we also characterize the bounds of these coefficients. These bounds implicitly encompass the existing results of Chan and Lam on trees and bipartite graphs. Furthermore, we give a solution to the open problem posed by Merris. Second, we characterize the first six coefficients of the hook immanantal polynomial. And the necessary and sufficient condition under which the linear combination matrices of two regular graphs have the same hook immanantal polynomial is proved. Third, we generalize the Frobenius--König theorem and the Laplace expansion theorem to immanants. Using these two theorems, we show that the star degree of a graph is always a lower bound for the multiplicity of a certain root of the immanantal polynomial of its linear combination matrix. Finally, we derive formulas for the first six coefficients of the hook immanantal polynomial for several important graph matrices.

math.CO↗

arXiv2026

Bollobás-type inequalities for subspaces via weight invariance

Let $V$ be an $n$-dimension real vector space with a direct sum decomposition $V = V_1 \oplus \cdots \oplus V_r$. Let $\mathcal{P} = \{(A_i, B_i) : i \in [m]\}$ be a skew Bollobás system of subspaces of $V$ such that each $i\in [m]$, $ A_i = \bigoplus_{k=1}^r (A_i \cap V_k)$ and $ B_i = \bigoplus_{k=1}^r (B_i \cap V_k)$. We prove that $$\sum_{i=1}^{m} \prod_{k=1}^{r} \left[ \binom{a_{i,k} + b_{i,k}}{a_{i,k}} (1 + a_{i,k} + b_{i,k})^{-1} \right] \leq 1,$$ where $a_{i,k} = \dim(A_i \cap V_k)$ and $b_{i,k} = \dim(B_i \cap V_k)$. This extends a recent result of Yue from set systems to finite dimensional subspaces. We then consider Tuza's theorem on weak Bollobás system for $d$-tuples. We give an alternative proof of the original set version of Tuza, and also establish its vector space analogue. Precisely, let $\mathcal{P} = \{(A_i^{(1)}, \ldots, A_i^{(d)}) : i \in [m]\}$ be a skew Bollobás system of $d$-tuples of subspaces of finite dimensional space $V$ with $a^{(\ell)}_i=\dim (A_i^{(\ell)})$. Then, for any positive real numbers $p_1, \ldots, p_d$ satisfying $p_1 + \cdots + p_d = 1$, we prove that $ \sum_{i=1}^{m} \prod_{\ell=1}^{d} p_{\ell}^{a_i^{(\ell)}} \leq 1. $

math.CO↗

arXiv2026

On complexity of substructure connectivity and restricted connectivity of graphs

The connectivity of a graph is an important parameter to evaluate its reliability. $k$-restricted connectivity (resp. $R^h$-restricted connectivity) of a graph $G$ is the minimum cardinality of a set $S$ of vertices in $G$, if exists, whose deletion disconnects $G$ and leaves each component of $G-S$ with more than $k$ vertices (resp. $δ(G-S)\geq h$). In contrast, structure (substructure) connectivity of $G$ is defined as the minimum number of vertex-disjoint subgraphs whose deletion disconnects $G$. As generalizations of the concept of connectivity, structure (substructure) connectivity, restricted connectivity and $R^h$-restricted connectivity have been extensively studied from the combinatorial point of view. Very little is known about the computational complexity of these variants, except for the recently established NP-completeness of $k$-restricted edge-connectivity. In this paper, we prove that the problems of determining structure, substructure, restricted, and $R^h$-restricted connectivity are all NP-complete.

cs.CC↗

arXiv2025

The number of rooted spanning forests of bicirculant graphs

A bi-Cayley graph over the cyclic group $(\mathbb{Z}_n, +)$ is called a bicirculant graph. Let $Γ=BC(\mathbb{Z}_n; R,T,S)$ be a bicirculant graph with $R=-R\subseteq \mathbb{Z}_n\setminus \{0\}$ and $T={-}T\subseteq \mathbb{Z}_n\setminus \{0\}$ and $S\subseteq \mathbb{Z}_n$. In this paper, using Chebyshev polynomials, we obtain a closed formula for the number of rooted spanning forests of $Γ$. Moreover, we investigate some arithmetic properties of the number of rooted spanning forests of $Γ$, and find its asymptotic behaviour as $n$ tends infinity.

math.CO↗

arXiv2025

Hook immanantal equalities for linear combination matrices of (di)graphs and their applications

Let $χ_λ$ be an irreducible character of the symmetric group $S_n$. For an $n \times n$ matrix $M = (m_{ij})$, define the immanant of $M$ corresponding to $χ_λ$ by \begin{eqnarray*} d_λ(M) = \sum_{σ\in S_n} χ_λ(σ) \prod_{i=1}^n m_{iσ(i)}. \end{eqnarray*} For $λ= (k, 1^{n-k})$, the immanant $d_{(k, 1^{n-k})}(M)$ is called the hook immanant and denoted by $d_k(M)$. The hook immanant polynomial of matrix $M$ is defined as $d_{k}(xI_n - M)$, where $I_n$ is the $n \times n$ identity matrix. Let $G$ and $\overrightarrow{G}$ be a graph and a digraph, respectively. Suppose that $D(G)$ and $A(G)$ (resp. $D(\overrightarrow{G})$ and $A(\overrightarrow{G})$) are the degree matrix and adjacency matrix of $G$ (resp. $\overrightarrow{G}$), respectively. In this paper, we characterize two hook immanantal equalities for the linear combination of matrices $βD(G)+γA(G)$ and $βD(\overrightarrow{G})+γA(\overrightarrow{G})$, where $β$ and $γ$ are real numbers. As applications, we derive recursive formulas for the hook immanantal polynomials and hook immanants of graph matrices.

math.CO↗

arXiv2025

Some new operated Lie polynomial identities and Gröbner-Shirshov bases

Bremner and Elgendy developed a classification of operated polynomial identities for linear operators on associative algebras, encompassing both classical and newly discovered cases. Within the framework of Rota's Program, each of these new operated associative polynomial identities was shown to be Gröbner-Shirshov. This naturally led to a question posed by Guo and collaborators: is each corresponding operated Lie polynomial identity also Gröbner-Shirshov? In this paper, we provide an affirmative answer by proving that each such Lie analogue indeed is Gröbner-Shirshov, thereby enriching the development of Rota's Program on algebraic operators within the Lie algebraic setting.

math.RA↗

arXiv2025

On the edge reconstruction of the second immanantal polynomials of undirected graph and digraph

Let $M=(m_{ij})$ be an $n\times n$ matrix. The second immanant of matrix $M$ is defined by \begin{eqnarray*} d_{2}(M)=\sum_{σ\in S_{n}}χ_{2}(σ)\prod_{s=1}^{n}m_{sσ(s)}, \end{eqnarray*} where $χ_{2}$ is the irreducible character of $S_{n}$ corresponding to the partition $(2^{1},1^{n-2})$. The polynomial $d_{2}(xI-M)$ is called the second immanantal polynomial of matrix $M$. Denote by $D(G)$ (resp. $D(\overrightarrow{G})$) and $A(G)$ (resp. $A(\overrightarrow{G})$) the diagonal matrix of vertex degrees and the adjacency matrix of undirected graph $G$ (resp. digraph $\overrightarrow{G}$), respectively. In this article, we prove that $d_{2}(xI-A(G))$ (resp. $d_{2}(xI-A(\overrightarrow{G}))$) can be reconstructed from the second immanantal polynomials of the adjacency matrix of all subgraphs in $\{G-uv,G-u-v|uv\in E(G)\}$ (resp. $\{\overrightarrow{G}-e|e\in E(\overrightarrow{G})\}$). Furthermore, the polynomial $d_{2}(xI-D(\overrightarrow{G})\pm A(\overrightarrow{G}))$ can also be reconstructed by the second immanantal polynomials of the (signless) Laplacian matrixs of all subgraphs in $\{\overrightarrow{G}-e|e\in E(\overrightarrow{G})\}$, respectively.

math.CO↗

arXiv2025

Improved bounds for the coefficient of flow polynomials

Let $G$ be a connected bridgeless $(n,m)$-graph which may have loops and multiedges, and let $F(G,t)$ denote the flow polynomial of $G$. Dong and Koh \cite{Dong1} established an upper bound for the absolute value of coefficient $c_{i}$ of $t^{i}$ in the expansion of $F(G,t)$, where $0\leqslant i \leqslant m-n+1$. In this paper, we refine the aforementioned bound. Specifically, we demonstrate that when $n \leqslant m \leqslant n+3$, $|c_{i}|\leqslant d_{i}$, where $d_{i}$ is the coefficient of $t^{i}$ in the expansion $\prod\limits_{j=1}^{m-n+1}(t+j)$; and when $m\geqslant n+4$, $|c_{i}|\leqslant d_{i}$, with $d_{i}$ being the coefficient of $t^{i}$ in the expansion $(t+1)(t+2)(t+3)^{2}(t+4)^{m-n-3}$. Furthermore, we prove that if $G$ is a connected bridgeless cubic graph having only real flow roots, then $b_{i}\leqslant |c_{i}|$, where $b_{i}$ is the coefficient of $t^{i}$ in the expansion $(t+1)(t+2)^{\frac{n}{2}}$. Notably, if $G$ is simple connected bridgeless cubic graph with only real flow roots, then $b_{i}$ is the coefficient of $t^{i}$ in the expansion $(t+1)(t+2)^{\frac{n}{2}-2}(t+3)^{2}$.

math.CO↗

arXiv2025

On the edge reconstruction of the second immanantal polynomials of undirected graph and digraph

math.CO↗

arXiv2025

Brualdi-Goldwasser-Michael problem for maximum permanents of {\rm(0,1)}-matrices

Let $\mathscr{U}(n,τ)$ be the set of all {\rm(0,1)}-matrices of order $n$ with exactly $τ$ 0's. Brualdi et al. investigated the maximum permanents of all matrices in $\mathscr{U}(n,τ)$(R.A. Brualdi, J.L. Goldwasser, T.S. Michael, Maximum permanents of matrices of zeros and ones, J. Combin. Theory Ser. A 47 (1988) 207--245.). And they put forward an open problem to characterize the maximum permanents among all matrices in $\mathscr{U}(n,τ)$. In this paper, we focus on the problem. And we characterize the maximum permanents of all matrices in $\mathscr{U}(n,τ)$ when $n^{2}-3n\leqτ\leq n^{2}-2n-1$. Furthermore, we also prove the maximum permanents of all matrices in $\mathscr{U}(n,τ)$ when $σ-kn\equiv0 (mod~k+1)$ and $(k+1)n-σ\equiv0(mod~k)$, where $σ=n^{2}-τ$, $kn\leqσ\leq (k+1)n$ and $k$ is integer.

math.CO↗

arXiv2024

Free weighted differential ($q$-tri)dendriform algebras

In the present paper, we propose the concepts of weighted differential ($q$-tri)dendriform algebras and give some basic properties of them. The corresponding free objects are constructed, in both the commutative and noncommutative contexts.

math.RA↗

arXiv2024

Weighted differential ($q$-tri)dendriform algebras

In this paper, we first introduce a weighted derivation on algebras over an operad $\cal P$, and prove that for the free $\cal P$-algebra, its weighted derivation is determined by the restriction on the generators. As applications, we propose the concept of weighted differential ($q$-tri)dendriform algebras and study some basic properties of them. Then Novikov-(tri)dendriform algebras are initiated, which can be induced from differential ($q$-tri) dendriform of weight zero. Finally, the corresponding free objects are constructed, in both the commutative and noncommutative contexts.

math.RA↗

arXiv2023

Further results on the permanental sums of bicyclic graphs

Let $G$ be a graph, and let $A(G)$ be the adjacency matrix of $G$. The permanental polynomial of $G$ is defined as $π(G,x)=\mathrm{per}(xI-A(G))$. The permanental sum of $G$ can be defined as the sum of absolute value of coefficients of $π(G,x)$. Computing the permanental sum is $\#$P-complete. Any a bicyclic graph can be generated from three types of induced subgraphs. In this paper, we determine the upper bound of permanental sums of bicyclic graphs generated from each a type of induced subgraph. And we also determine the second maximal permanental sum of all bicyclic graphs.

math.CO↗