arXiv subjects

Mingqing Zhai

Publications and source records attributed to Mingqing Zhai.

At least 19 recordsLinked to original sources

arXiv2026

Dense-core approach to the Brualdi--Hoffman--Tur\'{a}n problem on odd wheels

We present a unified presentation of the fixed-size adjacency-spectral extremal problem for odd wheels $W_{2k+1}$, where $k\geq2$ and $W_{2k+1}=K_1\vee C_{2k}$. The exceptional case $W_5$ and the general case $W_{2k+1}$, $k\ge3$, share the same dense-core reduction and edge-spectral stability, but have different rigidity structures. We prove that every $W_5$-free graph of sufficiently large size $m$ satisfies $\rho(G)^2-\rho(G)\le m,$ with equality precisely for $K_{n,n}$ with a perfect matching embedded in each part, where $n$ is even and $m=n^2+n$. For any fixed $k\ge3$, every $W_{2k+1}$-free graph of sufficiently large size $m$ satisfies $\rho(G)^2-(k-1)\rho(G)\le m-\binom{k}{2},$ with equality precisely for $K_k\vee qK_1$ when $m=\binom{k}{2}+kq$. Our results completely settle a conjecture proposed by Yu, Li and Peng and, via a distinct approach, further strengthen known results concerning odd cycles, friendship graphs and odd fan graphs for sufficiently large $m.$ The proof combines the edge-spectral stability theorem, residual functions and the dense-core method.

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arXiv2026

Edge-spectral supersaturation for tripartite color-critical graphs

We study edge-spectral supersaturation for two families of color-critical graphs with chromatic number three. For an integer $r\geq 1$, we define the spectral threshold \[ g_r(m):=\frac{r-1+\sqrt{4m-r^2+1}}{2}, \] which is the tight upper bound on the spectral radius of graphs avoiding $K_{s,t}^+$ (when $t+1\geq s\geq 3$) and $C_{2k+1}$ (when $r=k$), realized by split-graph constructions. First, let $t+1 \geq s\geq 3$ be fixed integers, and let $K_{s,t}^{+}$ be obtained by adding an edge to the part of size $s$ in $K_{s,t}$. We prove that every sufficiently large $m$-edge graph $G$ with $\rho(G)>g_{s-1}(m)$ contains $\Omega(m^{(s+t-1)/2})$ copies of $K_{s,t}^{+}$. Second, for any fixed $k\geq 2$, the condition $\rho(G)>g_k(m)$ forces $N(C_{2k+1},G)=\Omega(m^k).$ We also construct graphs showing that both lower bounds are tight up to constant factors. These results establish that exceeding the tight spectral Tur\'{a}n threshold $g_r(m)$ forces not just a single copy, but the optimal polynomial number of copies of these color-critical graphs. Thus, crossing the relevant split-graph spectral threshold forces the optimal polynomial order of copies, extending edge-spectral existence theorems to supersaturation results in the delicate three-chromatic regime.

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arXiv2026

Nikiforov's spectral consecutive cycle problem and the connected-matching method

Let $\rho(G)$ denote the adjacency spectral radius of a graph $G$ of order $n$. We determine the sharp constant in an open problem of Nikiforov (2008) on cycles of consecutive lengths. For every $\varepsilon>0$ and all sufficiently large $n$, if $G$ is an $n$-vertex graph with $\rho(G)>\sqrt{\lfloor{n^2/4}\rfloor},$ then $G$ contains a cycle $C_\ell$ for every integer length $3\le \ell\le (\frac{3-\sqrt5}{2}-\varepsilon)n.$ The constant $(3-\sqrt5)/2$ is best possible, as shown by the split graph $K_k\vee\overline K_{n-k}$ with $k\sim(3-\sqrt5)n/4$. Our result improves all previous results [LAA2008, CPC2020, JGT2023, JGT2023, GC2024]. The proof combines the degree form of Szemer\'edi's regularity lemma, a spectral matching theorem of Feng-Yu-Zhang, Weyl's inequality, a refinement of \L{}uczak's connected-matching embedding method, and other ideas.

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arXiv2026

A spectral threshold for triangle counting

The 1970 spectral extension of Mantel's theorem, proved by Nosal, states that every graph with $m$ edges and spectral radius $\rho_1>\sqrt{m}$ contains at least one triangle. Its quantitative refinement by Ning and Zhai later established that any graph $G$ with $m$ edges and spectral radius $\rho_1\geq\sqrt{m}$ contains at least $\lfloor\frac{\sqrt{m}-1}{2}\rfloor$ triangles, unless $G$ is a complete bipartite graph. In this paper, we further investigate the minimum number of triangles guaranteed under the strengthened spectral condition $\rho_1\geq\sqrt{m}+c$, where $c$ is a positive constant. We prove that for any constant $c\in (0,\frac{1}{2}]$ and all sufficiently large $m$, if $s=s(m)$ is a real-valued function satisfying $\lim_{m\to\infty} \frac{s}{m}=c$, then every $m$-edge graph $G$ with spectral radius $\rho_1$ satisfying $\rho_1^2\geq m-1+\frac{2s}{\rho_1-1}$ contains at least $s$ triangles. Moreover, we characterize the extremal graph achieving the minimal number of triangles. In particular, when $s=\frac{m-1}2$, our result settles a conjecture proposed by Li, Feng, and Peng.

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arXiv2026

Counting color-critical subgraphs under Nikiforov's condition

For a graph $G$ with $m$ edges, let $\rho(G)$ be its spectral radius, and let $N_F(G)$ denote the number of copies of $F$ in $G$. Nikiforov [Combin. Probab.\,Comput., 2002] proved that for $r\geq 2$, if $\rho(G)>\sqrt{(1-1/r)2m}$, then $N_{K_{r+1}}(G)\geq 1$. Furthermore, Bollob\'{a}s and Nikiforov [J. Combin. Theory, Ser. B, 2007] used $\rho(G)$ to establish a counting inequality for complete subgraphs. In this paper, we generalize and strengthen the above results to any color-critical graph $F$ with chromatic number at least four. More precisely, we demonstrated that under Nikiforov's condition, the number of copies of $F$ in $G$ satisfies $N_F(G)\geq\big(\gamma_F-o(1)\big)m^{(|F|-2)/2},$ where both the leading item and the constant $\gamma_F$ are optimal. Let $F$ be a non-star graph with $\chi(F)=r+1$, and let $G$ be any graph of sufficiently large size $m$ satisfying $N_F(G)=o(m^{|F|/2})$. To support the aforementioned counting arguments, we initially employ the method of progressive induction to tackle spectral problems, proving that $\rho(G)\leq\sqrt{(1-1/r+o(1))2m}$ for $r\geq 3$, and $\rho(G)\leq\sqrt{(1+o(1))m}$ for $r\in \{1,2\}$. Furthermore, we establish a stability result for edge-spectral supersaturation: specifically, if $r\geq 3$ and $\rho(G)\geq\sqrt{(1-1/r-o(1))2m}$, then $G$ differs from an $r$-partite Tur\'{a}n graph by $o(m)$ edges; if $r\in \{1,2\}$ and $\rho(G)\geq\sqrt{(1-o(1))m}$, then $G$ differs from a complete bipartite graph by $o(m)$ edges. This implies the well-known Erdos-Simonovits stability theorem and existing spectral stability theorems, by strengthening the setting from $F$-free graphs to graphs containing only a limited number of copies of $F$. Finally, we propose several counting-related open problems for further investigation.

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arXiv2026

Spectral extremal graphs on closed surfaces of fixed Euler genus

Graph theory on surfaces extends classical graph structures to topological surfaces, providing a theoretical foundation for characterizing the embedding properties of complex networks in constrained spaces. The study of bounding the spectral radius $\rho(G)$ of graphs on surfaces has a rich history that dates back to the 1990s. In this paper, we establish tight bounds for graphs of order $n$ that are embeddable on a surface with Euler genus $\gamma$. Specifically, if graph $G$ achieves the maximum spectral radius, then \begin{equation*} \begin{array}{ll} \frac32\!+\!\sqrt{2n\!-\!\frac{15}4}\!+\!\frac{3\gamma\!-\!1}{n}<\rho(G)<\frac32\!+\!\sqrt{2n\!-\!\frac{15}4}\!+\!\frac{3\gamma\!-\!0.95}{n}, \end{array} \end{equation*} which improves upon the earlier bound $\rho(G)\leq2+\sqrt{2n+8\gamma-6}$ by Ellingham and Zha [JCTB, 2000]. Furthermore, we prove that any extremal graph is obtained from $K_2 \nabla P_{n-2}$ by adding exactly $3\gamma$ edges, where `$\nabla$' means the join product. As a corollary, for $\gamma = 0$ and $n \geq 4.5 \times 10^6$, the graph $K_2 \nabla P_{n-2}$ is the unique planar extremal graph, thereby confirming a long-standing conjecture resolved by Tait and Tobin [JCTB, 2017]. Let $K_r^n$ be the graph of order $n$ obtained by attaching two paths of nearly equal length to two distinct vertices of $K_r$. Integrating spectral techniques with considerable structural analysis on surface graphs, we further derive the following sharp bounds: $\rho(G) \leq \rho(K_2 \nabla K_4^{n-2})$ for projective-planar graphs, and $\rho(G) \leq \rho(K_2 \nabla K_5^{n-2})$ for toroidal graphs. Our study presents a novel framework for exploring the eigenvalue-extremal problem on surface graphs with high Euler genus.

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arXiv2026

Advances on two spectral conjectures regarding booksize of graphs

The booksize $ \mathrm{bk}(G) $ of a graph $ G $, introduced by Erd\H{o}s, refers to the maximum integer $ r $ for which $G$ contains the book $ B_r $ as a subgraph. This paper investigates two open problems in spectral graph theory related to the booksize of graphs. First, we prove that for any positive integer $r$ and any $ B_{r+1} $-free graph $ G $ with $ m \geq (9r)^2 $ edges, the spectral radius satisfies $ \rho(G) \leq \sqrt{m} $. Equality holds if and only if $ G $ is a complete bipartite graph. This result improves the lower bound on the booksize of Nosal graphs (i.e., graphs with $ \rho(G) > \sqrt{m} $) from the previously established $ \mathrm{bk}(G) > \frac{1}{144}\sqrt{m} $ to $ \mathrm{bk}(G) > \frac{1}{9}\sqrt{m} $, presenting a significant advancement in the booksize conjecture proposed Li, Liu, and Zhang. Second, we show that for any positive integer $r$ and any non-bipartite $ B_{r+1} $-free graph $ G $ with $ m \geq (240r)^2 $ edges, the spectral radius $\rho$ satisfies $\rho^2<m-1+\frac{2}{\rho-1}$, unless $G$ is isomorphic to $S^+_{m,s}$ for some $s\in\{1,\ldots,r\}$. This resolves Liu and Miao's conjecture and further reveals an interesting phenomenon: even with a weaker spectral condition, $\rho^2\geq m-1+\frac2{\rho-1}$, we can still derive the supersaturation of the booksize for non-bipartite graphs.

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arXiv2025

The spectral Tur\'{a}n problem: Characterizing spectral-consistent graphs

Let ${\rm EX}(n,H)$ and ${\rm SPEX}(n,H)$ denote the families of $n$-vertex $H$-free graphs with the maximum size and the maximum spectral radius, respectively. A graph $H$ is said to be spectral-consistent if ${\rm SPEX}(n,H)\subseteq {\rm EX}(n,H)$ for sufficiently large $n$. A fundamental problem in spectral extremal graph theory is to determine which graphs are spectral-consistent. Cioab\u{a}, Desai and Tait [European J. Combin. 99 (2022) 103420] proposed the following conjecture: Let $H$ be any graph such that the graphs in ${\rm EX}(n,H)$ are Tur\'{a}n graph plus $O(1)$ edges. Then $H$ is spectral-consistent. Wang, Kang and Xue [J. Combin. Theory Ser. B 159 (2023) 20--41] confirmed this conjecture, along with a stronger result. Recently, Liu and Ning raised a general problem in spectral extremal graph theory: Characterize all graphs that are spectral-consistent. In this paper, we establish that for any finite graph $H$, if its decomposition family is matching-good, then $H$ is necessarily spectral-consistent. Notably, this structural condition is strictly weaker than the condition for spectral-consistency established by Wang, Kang, and Xue in their earlier work, thereby broadening the class of graphs known to satisfy the spectral-consistency property. Our main result enables us to fully characterize the spectral-consistency for several important families of forbidden graphs $H$, including generalized color-critical graphs, odd-ballooning of trees and complete bipartite graphs, as well as edge blow-up of non-bipartite graphs and certain special bipartite graphs. Furthermore, we present a streamlined proof for an existing spectral-consistency result due to Chen, Lei, and Li, simplifying their original argument. Finally, we propose several open problems to motivate future research in this area.

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arXiv2024

Spectral Radius of Graphs with Size Constraints: Resolving a Conjecture of Guiduli

We resolve a problem posed by Guiduli (1996) on the spectral radius of graphs satisfying the Hereditarily Bounded Property $P_{t,r}$, which requires that every subgraph $H$ with $|V(H)| \geq t$ satisfies $|E(H)| \leq t|V(H)| + r$. For an $n$-vertex graph $G$ satisfying $P_{t,r}$, where $t > 0$ and $r \geq -\binom{\lfloor t+1 \rfloor}{2}$, we prove that the spectral radius $\rho(G)$ is bounded above by $\rho(G) \leq c(s,t) + \sqrt{\lfloor t \rfloor n}$, where $s = \binom{\lfloor t \rfloor + 1}{2} + r$, thus affirmatively answering Guiduli's conjecture. Furthermore, we present a complete characterization of the extremal graphs that achieve this bound. These graphs are constructed as the join graph $K_{\lfloor t \rfloor} \nabla F$, where $F$ is either $K_3 \cup (n - \lfloor t \rfloor - 3)K_1$ or a forest consisting solely of star structures. The specific structure of such forests is meticulously characterized. Central to our analysis is the introduction of a novel potential function $\eta(F) = e(F) + (\lfloor t \rfloor - t)|V(F)|$, which quantifies the structural "positivity" of subgraphs. By combining edge-shifting operations with spectral radius maximization principles, we establish sharp bounds on $\eta^+(G)$, the cumulative positivity of $G$. Our results contribute to the understanding of spectral extremal problems under edge-density constraints and provide a framework for analyzing similar hereditary properties.

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arXiv2024

Extremal eigenvalues with respect to graph minors

Let $spex(n,H_{minor})$ denote the maximum spectral radius of $n$-vertex $H$-minor free graphs. The problem on determining this extremal value can be dated back to the early 1990s. Up to now, it has been solved for $n$ sufficiently large and some special minors, such as $\{K_{2,3},K_4\}$, $\{K_{3,3},K_5\}$, $K_r$ and $K_{s,t}$. In this paper, we find some unified phenomena on general minors. Every graph $G$ on $n$ vertices with spectral radius $\rho\geq spex(n,H_{minor})$ contains either an $H$ minor or a spanning book $K_{\gamma_H}\nabla(n-\gamma_H)K_1$, where $\gamma_H=|H|-\alpha(H)-1$. Furthermore, assume that $G$ is $H$-minor free and $\Gamma^*_s(H)$ is the family of $s$-vertex irreducible induced subgraphs of $H$, then $G$ minus its $\gamma_H$ dominating vertices is $\Gamma^*_{\alpha(H)+1}(H)$-minor saturate, and it is further edge-maximal if $\Gamma^*_{\alpha(H)+1}(H)$ is a connected family. As applications, we obtain some known results on minors mentioned above. We also determine the extremal values for some other minors, such as flowers, wheels, generalized books and complete multi-partite graphs. Our results extend some conjectures on planar graphs, outer-planar graphs and $K_{s,t}$-minor free graphs. To obtain the results, we combine stability method, spectral techniques and structural analyses. Especially, we give an exploration of using absorbing method in spectral extremal problems.

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arXiv2024

Turán numbers for non-bipartite graphs and applications to spectral extremal problems

Given a graph family $\mathcal{H}$ with $\min_{H\in \mathcal{H}}χ(H)=r+1\geq 3$. Let ${\rm ex}(n,\mathcal{H})$ and ${\rm spex}(n,\mathcal{H})$ be the maximum number of edges and the maximum spectral radius of the adjacency matrix over all $\mathcal{H}$-free graphs of order $n$, respectively. Denote by ${\rm EX}(n,\mathcal{H})$ (resp. ${\rm SPEX}(n,\mathcal{H})$) the set of extremal graphs with respect to ${\rm ex}(n,\mathcal{H})$ (resp. ${\rm spex}(n,\mathcal{H})$). In this paper, we use a decomposition family defined by Simonovits to give a characterization of which graph families $\mathcal{H}$ satisfy ${\rm ex}(n,\mathcal{H})<e(T_{n,r})+\lfloor \frac{n}{2r} \rfloor$. Furthermore, we completely determine ${\rm EX}\big(n,\mathbb{G}(F_1,\ldots,F_k)\big)$ for $n$ sufficiently large, where $\mathbb{G}(F_1,\ldots,F_k)$ denotes a finite graph family which consists of $k$ edge-disjoint $(r+1)$-chromatic color-critical graphs $F_1,\ldots,F_k$. This result strengthens a theorem of Győri, who settled the case that $F_1=\cdots =F_k = K_{r+1}$. Wang, Kang and Xue %[J. Combin. Theory Ser. B 159 (2023) 20--41] proved that ${\rm SPEX}(n,H)\subseteq {\rm EX}(n,H)$ for $n$ sufficiently large and any graph $H$ with ${\rm ex}(n,H)=e(T_{n,r})+O(1)$. As an application of our first theorem, we show that ${\rm SPEX}(n,\mathcal{H})\subseteq {\rm EX}(n,\mathcal{H})$ for $n$ sufficiently large and any finite family $\mathcal{H}$ with ${\rm ex}(n,\mathcal{H})<e(T_{n,r})+\lfloor \frac{n}{2r}\rfloor$. As an application of our second theorem we completely determine ${\rm SPEX}\big(n,\mathbb{G}(F_1,\ldots,F_k)\big)$ for $n$ sufficiently large. Finally, related problems are proposed for further research.

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arXiv2023

Signless Laplacian spectral radius of graphs without short cycles or long cycles

The signless Laplacian spectral radius of a graph $G$, denoted by $q(G)$, is the largest eigenvalue of its signless Laplacian matrix. In this paper, we investigate extremal signless Laplacian spectral radius for graphs without short cycles or long cycles. Let $\mathcal{G}(m,g)$ be the family of graphs on $m$ edges with girth $g$ and $\mathcal{H}(m,c)$ be the family of graphs on $m$ edges with circumference $c$. More precisely, we obtain the unique extremal graph with maximal $q(G)$ in $\mathcal{G}(m,g)$ and $\mathcal{H}(m,c)$, respectively.

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arXiv2023

Spectral extremal problem on $t$ copies of $\ell$-cycle

Denote by $tC_\ell$ the disjoint union of $t$ cycles of length $\ell$. Let $ex(n,F)$ and $spex(n,F)$ be the maximum size and spectral radius over all $n$-vertex $F$-free graphs, respectively. In this paper, we shall pay attention to the study of both $ex(n,tC_\ell)$ and $spex(n,tC_\ell)$. On the one hand, we determine $ex(n,tC_{2\ell+1})$ and characterize the extremal graph for any integers $t,\ell$ and $n\ge f(t,\ell)$, where $f(t,\ell)=O(t\ell^2)$. This generalizes the result on $ex(n,tC_3)$ of Erdős [Arch. Math. 13 (1962) 222--227] as well as the research on $ex(n,C_{2\ell+1})$ of Füredi and Gunderson [Combin. Probab. Comput. 24 (2015) 641--645]. On the other hand, we focus on the spectral Turán-type function $spex(n,tC_{\ell})$, and determine the extremal graph for any fixed $t,\ell$ and large enough $n$. Our results not only extend some classic spectral extremal results on triangles, quadrilaterals and general odd cycles due to Nikiforov, but also develop the famous spectral even cycle conjecture proposed by Nikiforov (2010) and confirmed by Cioabă, Desai and Tait (2022).

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arXiv2021

Counting substructures and eigenvalues II: quadrilaterals

Let $G$ be a graph and $\lambda(G)$ be the spectral radius of $G$. A previous result due to Nikiforov [Linear Algebra Appl., 2009] in spectral graph theory asserted that every graph $G$ on $m\geq 10$ edges contains a 4-cycle if $\lambda(G)>\sqrt{m}$. Define $f(m)$ to be the minimum number of copies of 4-cycles in such a graph. A consequence of a recent theorem due to Zhai et al. [European J. Combin., 2021] shows that $f(m)=\Omega(m)$. In this article, by somewhat different techniques, we prove that $f(m)=\Theta(m^2)$. We left the solution to $\lim\limits_{m\rightarrow \infty} \frac{f(m)}{m^2}$ as a problem, and also mention other ones for further study.

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arXiv2021

Counting substructures and eigenvalues I: triangles

Motivated by the counting results for color-critical subgraphs by Mubayi [Adv. Math., 2010], we study the phenomenon behind Mubayi's theorem from a spectral perspective and start up this problem with the fundamental case of triangles. We prove tight bounds on the number of copies of triangles in a graph with a prescribed number of vertices and edges and spectral radius. Let $n$ and $m$ be the order and size of a graph. Our results extend those of Nosal, who proved there is one triangle if the spectral radius is more than $\sqrt{m}$, and of Rademacher, who proved there are at least $\lfloor\frac{n}{2}\rfloor$ triangles if the number of edges is more than that of 2-partite Turán graph. These results, together with two spectral inequalities due to Bollobás and Nikiforov, can be seen as a solution to the case of triangles of a problem of finding spectral versions of Mubayi's theorem. In addition, we give a short proof of the following inequality due to Bollobás and Nikiforov [J. Combin. Theory Ser. B, 2007]: $t(G)\geq \frac{λ(G)(λ^2(G)-m)}{3}$ and characterize the extremal graphs. Some problems are proposed in the end.

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arXiv2021

Outerplanar Turán numbers of cycles and paths

A graph is outerplanar if it can be embedded in a plane such that all vertices lie on its outer face. The outerplanar Turán number of a given graph $H$, denoted by ${\rm ex}_{\mathcal{OP}}(n,H)$, is the maximum number of edges over all outerplanar graphs on $n$ vertices which do not contain a copy of $H$. In this paper, the outerplanar Turán numbers of cycles and paths are completely determined.

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arXiv2021

A strengthening of the spectral chromatic critical edge theorem: books and theta graphs

The chromatic critical edge theorem of Simonovits states that for a given color critical graph $H$ with $χ(H)=k+1$, there exists an $n_0(H)$ such that the Turán graph $T_{n,k}$ is the only extremal graph with respect to $ex(n,H)$ provided $n \geq n_0(H)$. Nikiforov's pioneer work on spectral graph theory implies that the color critical edge theorem also holds if $ex(n,H)$ is replaced by the maximum spectral radius and $n_0(H)$ is an exponential function of $|H|$. We want to know which color critical graphs $H$ satisfy that $n_0(H)$ is a linear function of $|H|$. Previous graphs include complete graphs and odd cycles. In this paper, we find two new classes of graphs: books and theta graphs. Namely, we prove that every graph on $n$ vertices with $ρ(G)>ρ(T_{n,2})$ contains a book of size greater than $\frac{n}{6.5}$. This can be seen as a spectral version of a 1962 conjecture by Erdős, which states that every graph on $n$ vertices with $e(G)>e(T_{n,2})$ contains a book of size greater than $\frac{n}{6}$. In addition, our result on theta graphs implies that if $G$ is a graph of order $n$ with $ρ(G)>ρ(T_{n,2})$, then $G$ contains a cycle of length $t$ for every $t\leq \frac{n}{7}$. This is related to an open question by Nikiforov which asks to determine the maximum $c$ such that every graph $G$ of large enough order $n$ with $ρ(G)>ρ(T_{n,2})$ contains a cycle of length $t$ for every $t\leq cn$.

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arXiv2021

Spectral radius, edge-disjoint cycles and cycles of the same length

In this paper, we give spectral conditions to guarantee the existence of two edge disjoint cycles and two cycles of the same length. These two results can be seen as spectral analogues of Erdős and Posa's size condition and Erdős' classic problem on non existence of two cycles of the same length. By using double leading eigenvectors skill, we further give spectral condition to guarantee the existence of $k$ edge disjoint triangles.

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