arXiv subjects

Shuchao Li

Publications and source records attributed to Shuchao Li.

At least 19 recordsLinked to original sources

arXiv2026

Anti-Ramsey Number for Suspension of Edge-Critical Graphs

An edge-colored graph is called a rainbow graph if all its edges have distinct colors. The \textit{anti-Ramsey number}, denoted by $\ar(n,F),$ for a fixed graph $F$ and a positive integer $n$, is the maximum number of colors used in an edge-coloring of the complete graph $K_n$ that contains no rainbow copy of $F$. Meanwhile, the \textit{Tur\'an number}, denoted by $\ex(n,F),$ for graph $F$ and $n$, is the maximum number of edges in an $n$-vertex graph that does not contain $F$ as a subgraph. For a vertex $v$ and a multiset $\mathcal{H}$ of graphs, the \textit{suspension} $\mathcal{H} + v$ of $\mathcal{H}$ is the graph obtained by connecting the vertex $v$ to all vertices of $H$ for each $H \in \mathcal{H}$. Let integers $k\ge 1$ and $r\ge 2$ be fixed, and suppose that $\mathcal{H}_{k+1}=\{H_1, H_2, \ldots, H_{k+1}\}+v$ satisfying $H_1, H_2, \ldots, H_{k+1}$ are pairwise vertex-disjoint edge-critical graphs, and $\chi(H_i)=r$ for $i=1,2,\ldots, k+1$.In this paper, we determine $ \ar(n,\mathcal{H}_{k+1}) $ for $k\ge 1$, $r\ge 2$ and sufficiently large $n$. This result unifies and generalizes a result of Liu et al. (arXiv:2411.08475) concerning the friendship graph, and a result of Lu et al. (arXiv:2507.13165) on the intersecting cliques.

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arXiv2026

Spectral extrema of 1-planar graphs with no short cycles or small cliques

The spectral Tur\'an type problem, initiated by Nikiforov in 2007, aims to determine the graphs among $n$-vertex $H$-free graphs having maximum spectral radius. In this paper, we study this problem for $1$-planar graphs, i.e., graphs that admit a drawing in the plane such that each edge is crossed at most once. Recently, Xu and Chang proved that the graphs among all $n$-vertex $K_5$-free $1$-planar graphs having maximum spectral radius lie within a small family of candidates. First, this paper explicitly identifies the unique spectral extremal graph among the $n$-vertex $K_5$-free $1$-planar graphs. Second, it establishes a structural reduction theorem: For any forbidden subgraph $F$ with $\delta(F)\ge2$ that is contained in $K_2\vee P_{n-2}^{2+}$ but not in $K_2\vee I_{n-2}$, every spectral extremal $F$-free $1$-planar graph contains a spanning complete bipartite graph $K_{2,n-2}$, where $P^{2+}_{n-2}$ is obtained from a path $u_1u_2\dots u_{n-2}$ by adding edge $u_1u_{n-2}$ and all edges $u_iu_{i+2}$ for $1\le i\le n-4$, and $I_{n-2}$ denotes the empty graph on $n-2$ vertices. As applications, the graph among all $n$-vertex $C_5$-free (resp. $2C_5$-free) $1$-planar graphs having maximum spectral radius is determined. These results extend spectral Tur\'{a}n type problems for $1$-planar graphs from cliques to cycles and their disjoint union.

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arXiv2026

Characterizing $A_α$-minimizer graphs: given order and independence number

For a given graph $ G $, let $ A(G) $, $ Q(G) $, and $ D(G) $ denote the adjacency matrix, signless Laplacian matrix, and diagonal degree matrix of $ G $, respectively. The $ A_α(G) $ matrix, proposed by Nikiforov, is defined as $ A_α(G)=αD(G)+(1 - α)A(G) $, where $ α\in[0,1] $. This matrix captures the gradual transition from $ A(G) $ to $ Q(G) $. Let $ \mathcal{G}_{n,γ} $ denote the family of all connected graphs with $ n $ vertices and independence number $ γ$. A graph in $ \mathcal{G}_{n,γ} $ is referred to as an $ A_α$-minimizer graph if it achieves the minimum $ A_α$ spectral radius. In this paper, we first demonstrate that the $ A_α$-minimizer graph in $ \mathcal{G}_{n,γ} $ must be a tree when $ γ\geq\left\lceil\frac{n}{2}\right\rceil $, and we provide several characterizations of such $ A_α$-minimizer graphs. We then specifically characterize the $ A_α$-minimizer graphs for the case $ γ= \left\lceil\frac{n}{2}\right\rceil + 1 $ when $n\geq 9$. Furthermore, we obtain a structural characterization for the $ A_α$-minimizer graph when $ γ=n - c $, where $ c\geq4 $ is an integer.

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arXiv2025

Spectral extrema of graphs with fixed size: forbidden a fan graph, friendship graph or theta graph

It is well-known that the Brualdi-Hoffman-Turán-type problem inquiries about the maximum spectral radius $ λ(G) $ of an $ F $-free graph $ G $ with $ m $ edges. Let $ θ_{1,p,q} $ denote the theta graph, which is constructed by connecting two vertices with 3 internally disjoint paths of lengths 1, $ p $, and $ q $ respectively. Let $ F_k $ be the fan graph, that is, the join of a $ K_1 $ and a path $ P_{k - 1} $. Let $ F_{k,3} $ be the friendship graph, obtained by having $ k $ triangles share a common vertex. In this paper, we utilize the $ k $-core method and spectral techniques to address some spectral extrema of graphs with a fixed number of edges. Firstly, we demonstrate that for $ m \geqslant \frac{9}{4}k^6 + 6k^5 + 46k^4 + 56k^3 + 196k^2 $ and $ k \geqslant 3 $, if $ G $ is $ F_{2k + 2} $-free, then $ λ(G) \leqslant \frac{k - 1 + \sqrt{4m - k^2 + 1}}{2} $. Equality holds if and only if $ G \cong K_k \vee (\frac{m}{k}-\frac{k - 1}{2})K_1 $. This validates a conjecture by Yu, Li, and Peng [Discrete Math. 348 (2025) 114391] and refines a recent result by Li, Zhai, and Shu [European J. Combin. 120 (2024) 103966]. Secondly, we show that for $ m \geqslant \frac{9}{4}k^6 + 6k^5 + 46k^4 + 56k^3 + 196k^2 $ with $ k \geqslant 3 $, if $ G $ is $ F_{k,3} $-free and has $ m $ edges, then $ λ(G) \leqslant \frac{k - 1 + \sqrt{4m - k^2 + 1}}{2} $. Equality holds precisely when $ G \cong K_k \vee (\frac{m}{k}-\frac{k - 1}{2})K_1 $. This confirms a conjecture put forward by Li, Lu, and Peng [Discrete Math. 346(2023)113680]. Finally, we identify the $ θ_{1,p,q} $-free graph with $ m $ edges that possesses the largest spectral radius, where $ q \geqslant p \geqslant 3 $ and $ p + q \geqslant 2k + 1 $. A further research problem is also proposed.

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arXiv2025

Spectral Turán-type problem in non-$r$-partite graphs: Forbidden generalized book graph $B_{r,k}$

Given a graph $H$, a graph is said to be $H$-free if it does not contain $H$ as a subgraph. A graph is color-critical when it has an edge whose removal leads to a reduction in its chromatic number. For a graph $H$ with a chromatic number of $r + 1$, we use $\text{spex}_{r + 1}(n, H)$ to represent the maximum spectral radius among non-$r$-partite $H$-free graphs of order $n$. The set of all non-$r$-partite $H$-free graphs of order $n$ that have a spectral radius of $\text{spex}_{r + 1}(n, H)$ is denoted as $\text{SPEX}_{r + 1}(n, H)$. For $r\geq2$ and $k\geq1$, we define $B_{r,k}$ as the graph constructed by connecting each vertex of $K_r$ to every vertex of an independent set with size $k$. We refer to $B_{r,k}$ as a book graph (in the case of $r = 2$) or a generalized book graph (when $r\geq3$). It should be noted that $B_{r,k}$ is a color-critical graph with a chromatic number of $r + 1$. Lin, Ning, and Wu (2021) identified the unique extremal graph within $\text{SPEX}_3(n, B_{2,1})$; Li and Peng (2023) determined the unique extremal graph in $\text{SPEX}_{r + 1}(n, B_{r,1})$ for all $r\geq2$. Quite recently, Liu and Miao (2025) specified the unique extremal graph in $\text{SPEX}_3(n, B_{2,k})$ for all $k\geq2$. Inspired by these remarkable results, this paper, relying on spectral stability theory, local structure characterization, along with the theory of characteristic equations and Rayleigh quotient equations, aims to determine the unique extremal graph in $\text{SPEX}_{r + 1}(n, B_{r,k})$ for $r\geq3$, $k\geq1$, and sufficiently large $n$. This work partially addresses an open problem put forward in [38].

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arXiv2025

The exact Turán number of generalized book graph $B_{r,k}$ in non-$r$-partite graphs

Given a graph $H,$ we say that a graph is \textit{$H$-free} if it does not contain $H$ as a subgraph. The Turán number $\ex(n,H)$ of $H$ is the maximum number of edges in an $n$-vertex $H$-free graph, the set of all the corresponding extremal graphs is denoted by $\Ex(n, H)$. The study of Turán number of graphs is a central topic in extremal graph theory. A graph is \textit{color-critical} if it contains an edge whose deletion reduces its chromatic number. Simonovits showed that if $H$ is a color-critical graph of chromatic number $r+1,$ then for sufficiently large $n,$ $\Ex(n, H)=\{T_r(n)\},$ the $r$-partite Turán graph of order $n.$ Given a color-critical graph $H$ with chromatic number $r+1,$ it is interesting to determine $H$-free non-$r$-partite graphs with maximum number of edges. For a graph $H$ with chromatic number $r+1,$ denote $\ex_{r+1}(n,H)$ the maximum number of edges in non-$r$-partite $H$-free graphs of order $n,$ the set of all non-$r$-partite $H$-free graphs of order $n$ and size $\ex_{r+1}(n,H)$ is denoted by $\Ex_{r+1}(n, H)$. For $r\geq 3,\,k\geq1,$ the generalized book graph ${B}_{r,k}$ is a graph obtained by joining every vertex of $K_r$ to every vertex of an independent set of size $k$. Note that ${B}_{r,k}$ is a color-critical graph of chromatic number $r+1.$ In this paper, based on the stability theory and local structure characterization, the exact value of $\ex_{r+1}(n,B_{r,k})$ is determined and all the corresponding extremal graphs are identified, where $r\geq 3,\,k\geq1$ and $n$ is sufficiently large.

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arXiv2024

Connected triangle-free planar graphs whose second largest eigenvalue is at most 1

Let $λ_2$ be the second largest eigenvalue of the adjacency matrix of a connected graph. In 2023, Li and Sun \cite{LiSun1} determined all the connected $\{K_{2,3}, K_4\}$-minor free graphs whose second largest eigenvalue $λ_2\le 1$. As a continuance of it, in this paper we completely identify all the connected $\{K_5,K_{3,3}\}$-minor free graphs without $C_3$ whose second largest eigenvalue does not exceed 1. This partially solves an open problem posed by Li and Sun \cite{LiSun1}: Characterize all connected planar graphs whose second largest eigenvalue is at most $1.$ Our main tools include the spectral theory and the local structure characterization of the planar graph with respect to its girth.

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arXiv2024

Spectral extremal results on the $A_α$-spectral radius of graphs without $K_{a,b}$-minor

An important theorem about the spectral Turán problem of $K_{a,b}$ was largely developed in separate papers. Recently it was completely resolved by Zhai and Lin [J. Comb. Theory, Ser. B 157 (2022) 184-215], which also confirms a conjecture proposed by Tait [J. Comb. Theory, Ser. A 166 (2019) 42-58]. Here, the prior work is fully stated, and then generalized with a self-contained proof. The more complete result is then used to better understand the relationship between the $A_α$-spectral radius and the structure of the corresponding extremal $K_{a,b}$-minor free graph.

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arXiv2024

Turán-type problems on $[a,b]$-factors of graphs, and beyond

Given a set of graphs $\mathcal{H}$, we say that a graph $G$ is \textit{$\mathcal{H}$-free} if it does not contain any member of $\mathcal{H}$ as a subgraph. Let $\text{ex}(n,\mathcal{H})$ (resp. $\text{ex}_{sp}(n,\mathcal{H})$) denote the maximum size (resp. spectral radius) of an $n$-vertex $\mathcal{H}$-free graph. Denote by $\text{Ex}(n, \mathcal{H})$ the set of all $n$-vertex $\mathcal{H}$-free graphs with $\text{ex}(n, \mathcal{H})$ edges. Similarly, let $\mathrm{Ex}_{sp}(n,\mathcal{H})$ be the set of all $n$-vertex $\mathcal{H}$-free graphs with spectral radius $\text{ex}_{sp}(n, \mathcal{H})$. For positive integers $a, b$ with $a\leqslant b$, an $[a,b]$-factor of a graph $G$ is a spanning subgraph $F$ of $G$ such that $a\leqslant d_F(v)\leqslant b$ for all $v\in V(G)$, where $d_F(v)$ denotes the degree of the vertex $v$ in $F.$ Let $\mathcal{F}_{a,b}$ be the set of all the $[a,b]$-factors of an $n$-vertex complete graph $K_n$. In this paper, we determine the Turán number $\text{ex}(n,\mathcal{F}_{a,b})$ and the spectral Turán number $\text{ex}_{sp}(n,\mathcal{F}_{a,b}),$ respectively. Furthermore, the bipartite analogue of $\text{ex}(n,\mathcal{F}_{a,b})$ (resp. $\text{ex}_{sp}(n,\mathcal{F}_{a,b})$) is also obtained. All the corresponding extremal graphs are identified. Consequently, one sees that $\mathrm{Ex}_{sp}(n,\mathcal{F}_{a,b})\subseteq \text{Ex}(n, \mathcal{F}_{a,b})$ holds for graphs and bipartite graphs. This partially answers an open problem proposed by Liu and Ning \cite{LN2023}. Our results may deduce a main result of Fan and Lin \cite{FL2022}.

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arXiv2024

On the $A_α$-index of graphs with given order and dissociation number

Given a graph $G,$ a subset of vertices is called a maximum dissociation set of $G$ if it induces a subgraph with vertex degree at most 1, and the subset has maximum cardinality. The cardinality of a maximum dissociation set is called the dissociation number of $G$. The adjacency matrix and the degree diagonal matrix of $G$ are denoted by $A(G)$ and $D(G),$ respectively. In 2017, Nikiforov proposed the $A_α$-matrix: $A_α(G)=αD(G)+(1-α)A(G),$ where $α\in[0,1].$ The largest eigenvalue of this novel matrix is called the $A_α$-index of $G.$ In this paper, we firstly determine the connected graph (resp. bipartite graph, tree) having the largest $A_α$-index over all connected graphs (resp. bipartite graphs, trees) with fixed order and dissociation number. Secondly, we describe the structure of all the $n$-vertex graphs having the minimum $A_α$-index with dissociation number $τ$, where $τ\geqslant\lceil\frac{2}{3}n\rceil.$ Finally, we identify all the connected $n$-vertex graphs with dissociation number $τ\in\{2,\lceil\frac{2}{3}n\rceil,n-1,n-2\}$ having the minimum $A_α$-index.

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arXiv2023

Thinking Like an Expert:Multimodal Hypergraph-of-Thought (HoT) Reasoning to boost Foundation Modals

Reasoning ability is one of the most crucial capabilities of a foundation model, signifying its capacity to address complex reasoning tasks. Chain-of-Thought (CoT) technique is widely regarded as one of the effective methods for enhancing the reasoning ability of foundation models and has garnered significant attention. However, the reasoning process of CoT is linear, step-by-step, similar to personal logical reasoning, suitable for solving general and slightly complicated problems. On the contrary, the thinking pattern of an expert owns two prominent characteristics that cannot be handled appropriately in CoT, i.e., high-order multi-hop reasoning and multimodal comparative judgement. Therefore, the core motivation of this paper is transcending CoT to construct a reasoning paradigm that can think like an expert. The hyperedge of a hypergraph could connect various vertices, making it naturally suitable for modelling high-order relationships. Inspired by this, this paper innovatively proposes a multimodal Hypergraph-of-Thought (HoT) reasoning paradigm, which enables the foundation models to possess the expert-level ability of high-order multi-hop reasoning and multimodal comparative judgement. Specifically, a textual hypergraph-of-thought is constructed utilizing triple as the primary thought to model higher-order relationships, and a hyperedge-of-thought is generated through multi-hop walking paths to achieve multi-hop inference. Furthermore, we devise a visual hypergraph-of-thought to interact with the textual hypergraph-of-thought via Cross-modal Co-Attention Graph Learning for multimodal comparative verification. Experimentations on the ScienceQA benchmark demonstrate the proposed HoT-based T5 outperforms CoT-based GPT3.5 and chatGPT, which is on par with CoT-based GPT4 with a lower model size.

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arXiv2023

Matching extension and matching exclusion via the size or the spectral radius of graphs

A graph $G$ is said to be $k$-extendable if every matching of size $k$ in $G$ can be extended to a perfect matching of $G$, where $k$ is a positive integer. We say $G$ is $1$-excludable if for every edge $e$ of $G$, there exists a perfect matching excluding $e$. In this paper, we first establish a lower bound on the size (resp. the spectral radius) of $G$ to guarantee that $G$ is $k$-extendable. Then we determine a lower bound on the size (resp. the spectral radius) of $G$ to guarantee that $G$ is $1$-excludable. All the corresponding extremal graphs are characterized.

math.CO↗

arXiv2023

Extensions on spectral extrema of $C_5/C_6$-free graphs with given size

Let $\mathcal{F}$ denote a set of graphs. A graph $G$ is said to be $\mathcal{F}$-free if it does not contain any element of $\mathcal{F}$ as a subgraph. The Turán number is the maximum possible number of edges in an $\mathcal{F}$-free graph with $n$ vertices. It is well known that classical Turán type extremal problem aims to study the Turán number of fixed graphs. In 2010, Nikiforov \cite{Nik2} proposed analogously a spectral Turán type problem which asks to determine the maximum spectral radius of an $\mathcal{F}$-free graph with $n$ vertices. It attracts much attention and many such problems remained elusive open even after serious attempts, and so they are considered as one of the most intriguing problems in spectral extremal graph theory. It is interesting to consider another spectral Turán type problem which asks to determine the maximum spectral radius of an $\mathcal{F}$-free graph with $m$ edges. Denote by $\mathcal{G}(m,\mathcal{F})$ the set of $\mathcal{F}$-free graphs with $m$ edges having no isolated vertices. Each of the graphs among $\mathcal{G}(m,\mathcal{F})$ having the largest spectral radius is called a maximal graph. Let $θ_{p,q,r}$ be a theta graph formed by connecting two distinct vertices with three independent paths of length $p,q$ and $r,$ respectively (length refers to the number of edges). In this paper, we firstly determine the unique maximal graph among $\mathcal{G}(m,θ_{1,2,3})$ and $\mathcal{G}(m,θ_{1,2,4}),$ respectively. Then we determine all the maximal graphs among $\mathcal{G}(m,C_5)$ (resp. $\mathcal{G}(m,C_6)$) excluding the book graph. These results extend some earlier results.

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arXiv2022

Sharp bounds on the $A_α$-index of graphs in terms of the independence number

Given a graph $G$, the adjacency matrix and degree diagonal matrix of $G$ are denoted by $A(G)$ and $D(G)$, respectively. In 2017, Nikiforov \cite{0007} proposed the $A_α$-matrix: $A_α(G)=αD(G)+(1-α)A(G),$ where $α\in [0, 1]$. The largest eigenvalue of this novel matrix is called the $A_α$-index of $G$. In this paper, we characterize the graphs with minimum $A_α$-index among $n$-vertex graphs with independence number $i$ for $α\in[0,1)$, where $i=1,\lfloor\frac{n}{2}\rfloor,\lceil\frac{n}{2}\rceil,{\lfloor\frac{n}{2}\rfloor+1},n-3,n-2,n-1,$ whereas for $i=2$ we consider the same problem for $α\in [0,\frac{3}{4}{]}.$ Furthermore, we determine the unique graph (resp. tree) on $n$ vertices with given independence number having the maximum $A_α$-index with $α\in[0,1)$, whereas for the $n$-vertex bipartite graphs with given independence number, we characterize the unique graph having the maximum $A_α$-index with $α\in[\frac{1}{2},1).$

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arXiv2022

Hermitian adjacency matrix of the second kind for mixed graphs

This contribution gives an extensive study on spectra of mixed graphs via its Hermitian adjacency matrix of the second kind { ($N$-matrix for short)} introduced by Mohar \cite{0001}. This matrix is indexed by the vertices of the mixed graph, and the entry corresponding to an arc from $u$ to $v$ is equal to the sixth root of unity $ω=\frac{1+{\bf i}\sqrt{3}}{2}$ (and its symmetric entry is $\barω=\frac{1-{\bf i}\sqrt{3}}{2}$); the entry corresponding to an undirected edge is equal to 1, and 0 otherwise. The main results of this paper include the following: {equivalent} conditions for a mixed graph that shares the same spectrum of its $N$-matrix with its underlying graph are given. A sharp upper bound on the spectral radius is established and the corresponding extremal mixed graphs are identified. Operations which are called two-way and three-way switchings are discussed--they give rise to some cospectral mixed graphs. We extract all the mixed graphs whose rank of its $N$-matrix is $2$ (resp. 3). Furthermore, we show that {if $M_G$ is a connected mixed graph with rank $2,$ then $M_G$ is switching equivalent to each connected mixed graph to which it is cospectral}. However, this does not hold for some connected mixed graphs with rank $3$. We identify all mixed graphs whose eigenvalues of its $N$-matrix lie in the range $(-α,\, α)$ for $α\in\left\{\sqrt{2},\,\sqrt{3},\,2\right\}$.

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arXiv2021

Adjacency eigenvalues of graphs without short odd cycles

It is well known that spectral Turán type problem is one of the most classical {problems} in graph theory. In this paper, we consider the spectral Turán type problem. Let $G$ be a graph and let $\mathcal{G}$ be a set of graphs, we say $G$ is \textit{$\mathcal{G}$-free} if $G$ does not contain any element of $\mathcal{G}$ as a subgraph. Denote by $λ_1$ and $λ_2$ the largest and the second largest eigenvalues of the adjacency matrix $A(G)$ of $G,$ respectively. In this paper we focus on the characterization of graphs without short odd cycles according to the adjacency eigenvalues of the graphs. Firstly, an upper bound on $λ_1^{2k}+λ_2^{2k}$ of $n$-vertex $\{C_3,C_5,\ldots,C_{2k+1}\}$-free graphs is established, where $k$ is a positive integer. All the corresponding extremal graphs are identified. Secondly, a sufficient condition for non-bipartite graphs containing an odd cycle of length at most $2k+1$ in terms of its spectral radius is given. At last, we characterize the unique graph having the maximum spectral radius among the set of $n$-vertex non-bipartite graphs with odd girth at least $2k+3,$ which solves an open problem proposed by Lin, Ning and Wu [Eigenvalues and triangles in graphs, Combin. Probab. Comput. 30 (2) (2021) 258-270].

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arXiv2021

HYPER^2: Hyperbolic Poincare Embedding for Hyper-Relational Link Prediction

Link Prediction, addressing the issue of completing KGs with missing facts, has been broadly studied. However, less light is shed on the ubiquitous hyper-relational KGs. Most existing hyper-relational KG embedding models still tear an n-ary fact into smaller tuples, neglecting the indecomposability of some n-ary facts. While other frameworks work for certain arity facts only or ignore the significance of primary triple. In this paper, we represent an n-ary fact as a whole, simultaneously keeping the integrity of n-ary fact and maintaining the vital role that the primary triple plays. In addition, we generalize hyperbolic Poincaré embedding from binary to arbitrary arity data, which has not been studied yet. To tackle the weak expressiveness and high complexity issue, we propose HYPER^2 which is qualified for capturing the interaction between entities within and beyond triple through information aggregation on the tangent space. Extensive experiments demonstrate HYPER^2 achieves superior performance to its translational and deep analogues, improving SOTA by up to 34.5\% with relatively few dimensions. Moreover, we study the side effect of literals and we theoretically and experimentally compare the computational complexity of HYPER^2 against several best performing baselines, HYPER^2 is 49-61 times quicker than its counterparts.

cs.CL↗

arXiv2021

An arithmetic criterion for graphs being determined by their generalized $A_α$-spectrum

Let $G$ be a graph on $n$ vertices, its adjacency matrix and degree diagonal matrix are denoted by $A(G)$ and $D(G)$, respectively. In 2017, Nikiforov \cite{0007} introduced the matrix $A_α(G)=αD(G)+(1-α)A(G)$ for $α\in [0, 1].$ The $A_α$-spectrum of a graph $G$ consists of all the eigenvalues (including the multiplicities) of $A_α(G).$ A graph $G$ is said to be determined by the generalized $A_α$-spectrum (or, DGA$_α$S for short) if whenever $H$ is a graph such that $H$ and $G$ share the same $A_α$-spectrum and so do their complements, then $H$ is isomorphic to $G$. In this paper, when $α$ is rational, we present a simple arithmetic condition for a graph being DGA$_α$S. More precisely, put $A_{c_α}:={c_α}A_α(G),$ here ${c_α}$ is the smallest positive integer such that $A_{c_α}$ is an integral matrix. Let $\tilde{W}_{α}(G)=\left[{\bf 1},\frac{A_{c_α}{\bf 1}}{c_α},\ldots, \frac{A_{c_α}^{n-1}{\bf 1}}{c_α}\right]$, where ${\bf 1}$ denotes the all-ones vector. We prove that if $\frac{\det \tilde{W}_{α}(G)}{2^{\lfloor\frac{n}{2}\rfloor}}$ is an odd and square-free integer and the rank of $\tilde{W}_{α}(G)$ is full over $\mathbb{F}_p$ for each odd prime divisor $p$ of $c_α$, then $G$ is DGA$_α$S except for even $n$ and odd $c_α\,(\geqslant 3)$. By our obtained results in this paper we may deduce the main results in \cite{0005} and \cite{0002}.

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