arXiv subjects

Dongxiu Cai

Publications and source records attributed to Dongxiu Cai.

4 recordsLinked to original sources

arXiv2026

Polynomial positivity cones for Coxeter roots and walks in trees

For a finite simple graph $G$ and an integer $k\ge0$, let $w_k(G)$ denote the number of walks of length $k$. We prove the conjecture of Täubig, Weihmann, Kosub, Hemmecke, and Mayr for every finite tree and determine all equality cases. If $T$ has $n\ge1$ vertices, then $n w_{k+1}(T)-2(n-1)w_k(T)\ge0$ for every $k\ge1$; for $n\ge3$, equality holds if and only if $T$ is a star and $k$ is even, whereas for $n=1$ or $n=2$, equality holds for every $k\ge1$. For non-Dynkin trees and even indices, the proof is based on a polynomial positivity cone associated with the adjacency operator of a finite graph and a positive real root of its simply-laced Coxeter system. For finite connected bipartite non-Dynkin graphs, we establish sufficient positivity conditions in terms of Coxeter orbits and inversion sets, and verify these conditions for indicator roots supported on connected induced subtrees. For non-Dynkin trees, this yields the rooted even-index inequality and, after summation, the corresponding global inequality. We also prove that if $G$ is a finite connected bipartite non-Dynkin simple graph, $\varnothing\ne U\subseteq V(G)$, and the subgraph of $G$ induced by $U$ is a tree, then $|U|w_{k+1}(G,U)-2(|U|-1)w_k(G,U)\ge0$ for every $k\ge0$, where $w_k(G,U)$ counts the length-$k$ walks in $G$ whose initial and terminal vertices lie in $U$; the intermediate vertices are unrestricted. The remaining even-index cases for finite Dynkin trees are handled by generating-function recurrences, while the odd-index cases follow from a spectral covariance identity.

math.CO↗

arXiv2026

The Equality Cases For the Laplacian Conjecture of Brouwer

The Laplacian conjecture of Brouwer asserts that for any graph $G$ of order n with $m$ edges, the sum of the $k$ largest Laplacian eigenvalues satisfies $s_k(G) \le m + \binom{k+1}{2}$ for $k=1, \ldots, n$. Later, Li and Guo in 2022 further proposed the full Brouwer's Laplacian spectrum conjecture. Recently, Kothari and Tudose in 2026 proved the Brouwer's conjecture. Motivated by their perfect proof and methods, we proved that for a simple graph of order $n$ with $m$ edges and $1\le k\le n-1$, $s_k(G) = m + \binom{k+1}{2}$ if and only if $G$ is a threshold graph with clique number $k+1$, which confirms the full Brouwer conjecture proposed by Li and Guo.

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arXiv2026

The Equality Cases for the Grone-Merris-Bai Theorem

The Grone--Merris inequality, conjectured by Grone and Merris~(1994) and first proved by Bai~(2011), states that for every graph $G$ of order $n$ and every $1\le k\le n$, $\sum_{i=1}^kλ_i(G)\le\sum_{i=1}^k d_i^*(G)$, where $λ_1\ge\cdots\geλ_n$ are the Laplacian eigenvalues and $d_1^*\ge\cdots\ge d_n^*$ is the conjugate degree sequence. In this paper we determine exactly when equality holds. Using the split-graph trace inequality developed by Kothari and Tudose~(2026) in their proof of Brouwer's Laplacian conjecture---which relies on Bai's theorem and also establishes the equivalence between the two conjectures---together with the recent characterization of the Brouwer equality cases by Cai, Chen, Yang and Zhang~(2027), we prove that equality holds in the Grone--Merris inequality if and only if the graph $G$ belongs to one of two explicitly described families. Both families are obtained from a threshold graph by a surgical operation at one terminal block: in the first family, edges are removed from the initial dominating block; in the second, edges are added inside the initial isolated block. Our analysis yields a complete combinatorial description of all pairs $(G,k)$ for which the Grone--Merris bound is tight.

math.CO↗

arXiv2026

Sharp Bounds for Guiduli-Type Hereditary Spectral Problems

Guiduli asked in 1996 the following problem concerning the maximum spectral radius of a graph under hereditary density constraints. If an $n$-vertex graph $G$ satisfies $e(H)\le c|V(H)|^2$ for every subgraph $H$ of $G$, must one have $λ(G)\le 2cn$? More generally, what remains true when the exponent $2$ is replaced by a constant less than $2$? We study the natural power-law version of this question for all $1<p\le2$. For $1<p\le 2$, define \[ d_p(G)=\max_{\varnothing\ne S\subseteq V(G)}\frac{e(G[S])}{|S|^p}. \] We determine the sharp asymptotic upper bound for $λ(G)$ in terms of $d_p(G)$ and $n$. More precisely, every $n$-vertex graph $G$ with at least one edge satisfies \[ λ(G)\le \begin{cases} \left(\left(\max_{t\in\mathbb N_{\ge1}}\dfrac{t}{(t+1)^p}\right)^{-1}+o(1)\right)d_p(G)\sqrt n,&1<p<3/2,\\[0.4em] \left(\dfrac{3\sqrt3}{4}+o(1)\right)d_p(G)\sqrt{n\log n},&p=3/2,\\[0.4em] (\mathfrak C_p+o(1))d_p(G)n^{p-1},&3/2<p<2, \end{cases} \] and each constant here is best possible. Here $\mathfrak C_p$ is characterized by an exact variational problem over finite kernels. We apply a sparse graphon operator estimate to convert hereditary $p$-density bounds into sharp spectral bounds, and this estimate also explains the transition at the critical exponent $p=3/2$. For the endpoint $p=2$, Wilf's theorem gives the exact finite-$n$ bound $λ(G)\le 2d_2(G)n$, with equality for $K_n$. Thus Guiduli's power-law problem is resolved in its sharp asymptotic form for every $1<p\leq2$, including exact leading constants.

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