Quanyu Tang

The number of Laplacian eigenvalues of trees less than one

Jiaxin Guo, Tao Hu, Quanyu Tang

Let $m_{T} [0, 1)$ denote the number of Laplacian eigenvalues of a tree $T$ that are strictly less than $1$ . Guo, Xue and Liu \cite{GXL} proved that every tree $T$ of diameter $d$ sati…

math.NT2026

Integral Representations and Asymptotics for a Family of Areal Mahler Measures

Quanyu Tang, Shu Zhang

We study a problem posed by Matilde Lalín concerning the areal Mahler measures of the multivariable polynomial family $$ P_m(x_1,\ldots,x_m,u) = \prod_{j=1}^m(1+x_j) + u\prod_{j=1}…

The $Δ$ -Conjecture for CIS $d$ -Graphs

Yinchen Liu, Quanyu Tang

We prove the $Δ$ -conjecture, which dates back to Gurvich's 1978 thesis. Specifically, let the edges of a complete graph be colored with colors $1, \dots, d$ , and for each $i$ let $G…

Nearly permanental cospectral graphs

Weifang Lv, Quanyu Tang, Wei Wang +1

Let $G$ be a simple graph of order $n$ with adjacency matrix $A = (a_{ij})$ . The \emph{determinant} and the \emph{permanen}t of the matrix $A$ are defined as \[\mathrm{det}A= \sum_{…

A Family of Simultaneously Cospectral Trees for Degree-Distance Matrices

Limeng Lin, Quanyu Tang, Kehua Wang +1

Spectral characterization of graphs for various graph matrices constitutes a central topic in spectral graph theory. Let $G$ be a graph with adjacency matrix $A (G)$ , diagonal degre…

Graph Eigenvalues and Projection Constants

Varun Sivashankar, Quanyu Tang, Tanay Wakhare

For an integer $k \geq 2$ , let $λ_{k} (G)$ denote the $k$ th largest adjacency eigenvalue of a graph $G$ . For every graph $G$ on $n$ vertices and every $2 \leq k \leq n$ , we prove \[ λ_k(…

Minimum-rank parameters of complements of threshold Kneser graphs

Tao Hu, Quanyu Tang

Let $J_{\geq s} (n, k)$ be the graph whose vertices are the $k$ -subsets of $[n]$ , with two distinct vertices adjacent whenever their intersection has size at least $s$ . Equivalently,…

A positive square-energy strengthening of Turán's theorem

Yinchen Liu, Quanyu Tang, Shengtong Zhang

Let $G$ be an $n$ -vertex graph with clique number $ω (G)$ , and let $s^{+} (G)$ denote the sum of the squared positive adjacency eigenvalues. We prove that $$ \sqrt{s^+(G)}\le\left(1-\f…

The positive and negative square-energy conjecture

Yinchen Liu, Quanyu Tang, Shengtong Zhang

Let $s^{+} (G)$ and $s^{-} (G)$ denote the sums of the squares of the positive and negative adjacency eigenvalues of a graph $G$ , respectively. We prove the conjecture of Elphick, Farber…

Irregular subgraph in a regular graph

Tianyue Cao, Quanyu Tang, Hehui Wu

A conjecture of Alon and Wei states that, for any $d$ -regular graph $G$ with $n$ vertices, there exists a spanning subgraph $H$ such that for all $0 \leq i \leq d$ , we have $m (H, i)$ ,…

Generalized spectral closedness of $F$ -free graph classes

Wei Wang, Quanyu Tang

In this paper, we investigate the generalized spectral closedness of graph classes defined by a family $F$ of forbidden induced subgraphs. To systematically study this pr…

Path-Minimality for Positive $p$ -Energies, Laplacian-Type Spectra, and Line Graphs

Yinchen Liu, Quanyu Tang

We derive several applications of the path-minimality theorem for adjacency $p$ -energy proved in the companion paper. First, we prove the sharp inequality $$ \mathcal E_p^+(G)\ge \…

Factorization of invariant polynomials and generalized spectral characterizations of graphs

Wei Wang, Quanyu Tang

The problem of characterizing graphs by their generalized spectra has received significant attention in recent years. This paper provides a complete proof of a conjecture proposed…

math.NT2026

The Hofstadter consecutive-sum sequence omits infinitely many positive integers

Quanyu Tang

Let $(a_{n})_{n \geq 1}$ be the greedy self-generating sequence defined by $a_{1} = 1$ , $a_{2} = 2$ , and, for $k \geq 3$ , by taking $a_{k}$ to be the least integer greater than $a_{k - 1}$ that can…

Proof of a conjectured spectral upper bound on the chromatic number of a graph

Quanyu Tang, Clive Elphick

Let $G$ be a simple graph on $n$ vertices and $m$ edges with chromatic number $χ$ , and let $λ_{n}$ denote the least adjacency eigenvalue. Solving a conjecture of Fan, Yu and Wang~[El…

Nordhaus--Gaddum type bounds for the complement rank

Quanyu Tang

Let $G$ be an $n$ -vertex simple graph with adjacency matrix $A_{G}$ . The complement rank of $G$ is defined as $rank (A_{G} + I)$ , where $I$ is the identity matrix. In this…

A new conjecture on the inertia of graphs

Saieed Akbari, Clive Elphick, Hitesh Kumar +2

Let $G$ be a graph with adjacency matrix $A (G)$ . We conjecture that \[2n^+(G) \le n^-(G)(n^-(G) + 1),\] where $n^{+} (G)$ and $n^{-} (G)$ denote the number of positive and negative eigen…

Inertia, Independence and Expanders

Quanyu Tang, Shengtong Zhang, Clive Elphick

Let $G$ be a graph on $n$ vertices, independence number $α (G)$ , Lovász theta function $ϑ (G)$ , and Shannon capacity $Θ (G)$ . We define $n_{\geq 0} (G)$ to be the minimum number…

A Spectral Lower Bound on Chromatic Numbers using $p$ -Energy

Clive Elphick, Quanyu Tang, Shengtong Zhang

Let $A_{G}$ be the adjacency matrix of a simple graph $G$ , and let $χ (G)$ , $χ_{f} (G)$ , $χ_{q} (G)$ , $ξ (G)$ and $ξ_{f} (G)$ denote its chromatic number, fractional chromatic numb…

New arithmetic invariants for cospectral graphs

Yizhe Ji, Quanyu Tang, Wei Wang +1

An invariant for cospectral graphs is a property shared by all cospectral graphs. In this paper, we establish three novel arithmetic invariants for cospectral graphs, revealing dee…

The Generation of All Regular Rational Orthogonal Matrices

Quanyu Tang, Wei Wang, Hao Zhang

A \emph{rational orthogonal matrix} $Q$ is an orthogonal matrix with rational entries, and $Q$ is called \emph{regular} if each of its row sum equals one, i.e., $Q e = e$ where $e$ …

On a conjecture of Nikiforov concerning the minimal $p$ -energy of connected graphs

Quanyu Tang, Yinchen Liu, Wei Wang

For a given simple graph $ G $, the $ p $-energy of $ G $, denoted by $ \mathcal{E}_p(G) $, is defined as the sum of the $ p $-th power of the absolute values of the eige…