arXiv subjects

Longfei Fang

Publications and source records attributed to Longfei Fang.

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.

math.CO↗

arXiv2026

Clique supersaturation under a chromatic constraint below the Tur\'{a}n threshold

A central theme in extremal graph theory is the supersaturation problem, which investigates the minimum number of copies of a target subgraph forced by prescribed edge conditions. This line of research goes back to Rademacher and Erd\H{o}s for triangles, and was later extended to cliques by Lov\'asz and Simonovits in the regime above the Tur\'an threshold. Mubayi further extended this theory to color-critical graphs. Below the Tur\'an threshold, a closely related existence-threshold phenomenon arises in the non-$p$-partite setting: a classical result of Brouwer shows that, for $n\ge 2p+1$, every $n$-vertex non-$p$-partite $K_{p+1}$-free graph has at most $e(T_{n,p})-\lfloor n/p\rfloor+1$ edges. Motivated by this threshold, we investigate a sharp clique-counting problem below the Tur\'an threshold under the non-$p$-partite assumption. Let $p\ge 2$ and $s\ge 1$ be fixed integers. Let $Y_{n,p,s}$ be the graph obtained from $T_{n,p}$ by adding an edge inside a largest part and deleting all but $s$ of the edges from one endpoint of this new edge to a smallest part. Then $e(Y_{n,p,s})=e(T_{n,p})-\lfloor n/p\rfloor+s+1$. We prove that, for all sufficiently large $n$, every $n$-vertex non-$p$-partite graph $G$ with $e(G)\ge e(Y_{n,p,s})$ contains at least as many copies of $K_{p+1}$ as $Y_{n,p,s}$ does. The bound is sharp, as it is attained by the construction $Y_{n,p,s}$. Thus our result provides the exact clique-counting analogue of Brouwer's threshold for non-$p$-partite $K_{p+1}$-free graphs.

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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

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

More on spectral supersaturation for the bowtie

A central topic in extremal graph theory is the supersaturation problem, which studies the minimum number of copies of a fixed substructure that must appear in any graph with more edges than the corresponding Tur\'an number. Significant works due to Erd\H{o}s, Rademacher, Lov\'{a}sz and Simonovits investigated the supersaturation problem for the triangle. Moreover, Kang, Makai and Pikhurko studied the case for the bowtie, which consists of two triangles sharing a vertex. Building upon the pivotal results established by Bollob\'{a}s, Nikiforov, Ning and Zhai on counting triangles via the spectral radius, we study in this paper the spectral supersaturation problem for the bowtie. Let $\lambda (G)$ be the spectral radius of a graph $G$, and let $K_{\lceil \frac{n}{2}\rceil, \lfloor \frac{n}{2}\rfloor}^q$ be the graph obtained from Tur\'{a}n graph $T_{n,2}$ by adding $q$ pairwise disjoint edges to the partite set of size $\lceil \frac{n}{2}\rceil$. Firstly, we prove that there exists an absolute constant $\delta >0$ such that if $n$ is sufficiently large, $2\le q \le \delta \sqrt{n}$, and $G$ is an $n$-vertex graph with $\lambda (G)\ge \lambda (K_{\lceil \frac{n}{2}\rceil, \lfloor \frac{n}{2}\rfloor}^q)$, then $G$ contains at least ${q\choose 2}\lfloor \frac{n}{2}\rfloor$ bowties, and $K_{\lceil \frac{n}{2}\rceil, \lfloor \frac{n}{2}\rfloor}^q$ is the unique spectral extremal graph. This solves an open problem proposed by Li, Feng and Peng. Secondly, we show that a graph $G$ whose spectral radius exceeds that of the spectral extremal graph for the bowtie must contain at least $\lfloor \frac{n-1}{2}\rfloor$ bowties. This sharp bound reveals a distinct phenomenon from the edge-supersaturation case, which guarantees at least $\lfloor \frac{n}{2}\rfloor$ bowties.

math.CO↗

arXiv2025

Spectral supersaturation for color-critical graphs

A graph is color-critical if it contains an edge whose deletion reduces its chromatic number. This class of graphs, including cliques and odd cycles, plays a central role in extremal graph theory. In this paper, following an influential line of research initiated by Bollob\'as-Nikiforov, we study the spectral supersaturation problem for color-critical graphs. Let $T_{n,r}$ be the $r$-partite Tur\'an graph, let $\mathcal{T}_{n,r,q}$ denote the family of graphs obtained from $T_{n,r}$ by adding $q$ edges, and let $\lambda(G)$ be the spectral radius of a graph $G$. We first prove that for any color-critical graph $F$ with chromatic number $r+1$, there exists $\delta_F > 0$ such that for sufficiently large $n$ and all $1 \leq q \leq \delta_F \sqrt{n}$, any $n$-vertex graph $G$ with $\lambda(G) \ge \min_{T \in \mathcal{T}_{n,r,q}} \lambda(T)$ contains at least $q \cdot c(n,F)$ copies of $F$, where $c(n,F)$ denotes the minimum number of copies of $F$ created by adding a single edge to $T_{n,r}$; moreover, any extremal graph $G$ must belong to $ \mathcal{T}_{n,r,q}$.Next, we prove a spectral supersaturation result for the analogous condition $\lambda(G) \ge \max_{T \in \mathcal{T}_{n,r,q}} \lambda(T)$, valid for all $1 \leq q \leq \delta_F n$. Together, these results provide a complete resolution to a problem proposed by Ning-Zhai, and establish a spectral counterpart to the well-known results of Mubayi and Pikhurko-Yilma in the extremal supersaturation setting. A notable feature of our first result is that the restriction $q = O(\sqrt{n})$ is tight up to a constant factor, in contrast to the linear bounds provided by other settings discussed above. As applications, we extend a result of Liu-Mubayi, and solve a related conjecture by Li-Lu-Peng.

math.CO↗

arXiv2025

On the Tur\'{a}n number of odd-ballooning of $3$-chromatic graphs

Given a graph $F$, the Tur\'{a}n number ${\rm ex}(n,F)$ is the maximum number of edges in any $n$-vertex $F$-free graph. The odd-ballooning of $F$, denoted by $F^{o}$, is a graph obtained by replacing each edge of $F$ with an odd cycle, where all new vertices of the odd cycles are distinct. The Tur\'{a}n number of the odd-ballooning of $F$ has been established for several important cases. For a star, it was determined by Erd\H{o}s, F\"{u}redi, Gould, and Gunderson (1995), Hou, Qiu, and Liu (2018), and Yuan (2018); for trees under certain conditions, by Zhu and Chen (2023); and for complete bipartite graphs $K_{s,t}$ ($t\geq s \geq 2$) where each substituted odd cycle has length at least five, by Peng and Xia (2024). In this paper, we apply Simonovits' celebrated method of progressive induction to determine the Tur\'{a}n number for the odd-ballooning of a class of $3$-chromatic graphs. Specifically, let $F$ be a graph formed by connecting a single vertex to all vertices of another graph whose components are either non-trivial trees or even cycles. We determine ${\rm ex}(n,F^{o})$ when each substituted odd cycle in $F^{o}$ has length at least five. As corollaries, we obtain the Tur\'{a}n number for the odd-ballooning of several well-known graph classes, including odd wheels, fan graphs, book graphs, and friendship graphs, where each substituted odd cycle in the ballooning has length at least five.

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arXiv2025

Non-bipartite graphs without theta subgraphs

Fix a color-critical graph $H$ with $\chi(H)=r+1\geq 3$. Simonovits' chromatic critical edge theorem and Nikiforov's spectral chromatic critical edge theorem imply that $T_{n,r}$ is the extremal graph with the maximum size and the maximum spectral radius over all $H$-free graphs of order $n$, respectively. Since $T_{n,r}$ is $r$-partite, it is interesting to study the Tur\'{a}n number and the spectral Tur\'{a}n number of a color-critical graph $H$ in non-$r$-partite graphs. Denote by ${\rm EX}_{r+1}(n,H)$ (resp. ${\rm SPEX}_{r+1}(n,H)$) the family of $n$-vertex $H$-free non-$r$-partite graphs with the maximum size (resp. spectral radius). Brouwer showed that any graph in $\mathrm{EX}_{r+1}(n,K_{r+1})$ is of size $e(T_{n,r})-\lfloor\frac{n}{r}\rfloor+1$ for $n\geq 2r+1$. Lin, Ning and Wu [Combin. Probab. Comput. 30 (2) (2021) 258--270], and Li and Peng [SIAM J. Discrete Math. 37 (2023) 2462--2485] characterized the unique graph in $\mathrm{SPEX}_{r+1}(n,K_{r+1})$ for $r\geq 2$. Particularly, the unique graph is of size $e(T_{n,r})-\lfloor\frac{n}{r}\rfloor+1$. Thus $\mathrm{SPEX}_{r+1}(n,K_{r+1})\subseteq \mathrm{EX}_{r+1}(n,K_{r+1})$. It is natural to conjecture that ${\rm SPEX}_{r+1}(n,H)\subseteq {\rm EX}_{r+1}(n,H)$ for arbitrary color-critical graph $H$ with $\chi(H)=r+1\geq 3$. Fix $q,r\geq 2$ with even $q$, a theta graph $\theta(1,q,r)$ is obtained from internally disjoint paths of lengths $1,q,r$, respectively by sharing a common pair of endpoints. In this paper, we prove that $\mathrm{SPEX}_{3}(n,\theta(1,q,r))\subseteq \mathrm{EX}_{3}(n,\theta(1,q,r))$ for sufficiently large $n$. Furthermore, we determine all the graphs in $\mathrm{SPEX}_{3}(n,\theta(1,q,r))$ and $\mathrm{EX}_{3}(n,\theta(1,q,r))$, respectively.

math.CO↗

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

Long cycles and spectral radii in planar graphs

There is a rich history of studying the existence of cycles in planar graphs. The famous Tutte theorem on the Hamilton cycle states that every 4-connected planar graph contains a Hamilton cycle. Later on, Thomassen (1983), Thomas and Yu (1994) and Sanders (1996) respectively proved that every 4-connected planar graph contains a cycle of length $n-1, n-2$ and $n-3$. Chen, Fan and Yu (2004) further conjectured that every 4-connected planar graph contains a cycle of length $\ell$ for $\ell\in\{n,n-1,\ldots,n-25\}$ and they verified that $\ell\in \{n-4, n-5, n-6\}$. When we remove the ``4-connected" condition, how to guarantee the existence of a long cycle in a planar graph? A natural question asks by adding a spectral radius condition: What is the smallest constant $C$ such that for sufficiently large $n$, every graph $G$ of order $n$ with spectral radius greater than $C$ contains a long cycle in a planar graph? In this paper, we give a stronger answer to the above question. Let $G$ be a planar graph with order $n\geq 1.8\times 10^{17}$ and $k\leq \lfloor\log_2(n-3)\rfloor-8$ be a non-negative integer, we show that if $\rho(G)\geq \rho(K_2\vee(P_{n-2k-4}\cup 2P_{k+1}))$ then $G$ contains a cycle of length $\ell$ for every $\ell\in \{n-k, n-k-1, \ldots, 3\}$ unless $G\cong K_2\vee(P_{n-2k-4}\cup 2P_{k+1})$.

math.CO↗

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.

math.CO↗

arXiv2024

Tur\'an numbers for non-bipartite graphs and applications to spectral extremal problems

Given a graph family $\mathcal{H}$ with $\min_{H\in \mathcal{H}}\chi(H)=r+1\geq 3$. 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 Gy\H{o}ri, 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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arXiv2024

Spectral extremal results on trees

Let ${\rm spex}(n,F)$ be the maximum spectral radius over all $F$-free graphs of order $n$, and ${\rm SPEX}(n,F)$ be the family of $F$-free graphs of order $n$ with spectral radius equal to ${\rm spex}(n,F)$. Given integers $n,k,p$ with $n>k>0$ and $0\leq p\leq \lfloor(n-k)/2\rfloor$, let $S_{n,k}^{p}$ be the graph obtained from $K_k\nabla(n-k)K_1$ by embedding $p$ independent edges within its independent set, where `$\nabla$' means the join product. For $n\geq\ell\geq 4$, let $G_{n,\ell}=S_{n,(\ell-2)/2}^{0}$ if $\ell$ is even, and $G_{n,\ell}=S_{n,(\ell-3)/2}^{1}$ if $\ell$ is odd. Cioab\u{a}, Desai and Tait [SIAM J. Discrete Math. 37 (3) (2023) 2228--2239] showed that for $\ell\geq 6$ and sufficiently large $n$, if $\rho(G)\geq \rho(G_{n,\ell})$, then $G$ contains all trees of order $\ell$ unless $G=G_{n,\ell}$. They further posed a problem to study ${\rm spex}(n,F)$ for various specific trees $F$. Fix a tree $F$ of order $\ell\geq 6$, let $A$ and $B$ be two partite sets of $F$ with $|A|\leq |B|$, and set $q=|A|-1$. We first show that any graph in ${\rm SPEX}(n,F)$ contains a spanning subgraph $K_{q,n-q}$ for $q\geq 1$ and sufficiently large $n$. Consequently, $\rho(K_{q,n-q})\leq {\rm spex}(n,F)\leq \rho(G_{n,\ell})$, we further respectively characterize all trees $F$ with these two equalities holding. Secondly, we characterize the spectral extremal graphs for some specific trees and provide asymptotic spectral extremal values of the remaining trees. In particular, we characterize the spectral extremal graphs for all spiders, surprisingly, the extremal graphs are not always the spanning subgraph of $G_{n,\ell}$.

math.CO↗

arXiv2023

Spectral extremal results on edge blow-up of graphs

Let ${\rm ex}(n,F)$ and ${\rm spex}(n,F)$ be the maximum size and maximum spectral radius of an $F$-free graph of order $n$, respectively. The value ${\rm spex}(n,F)$ is called the spectral extremal value of $F$. Nikiforov [J. Graph Theory 62 (2009) 362--368] gave the spectral Stability Lemma, which implies that for every $\varepsilon>0$, sufficiently large $n$ and a non-bipartite graph $H$ with chromatic number $\chi(H)$, the extremal graph for ${\rm spex}(n,H)$ can be obtained from the Tur\'{a}n graph $T_{\chi(H)-1}(n)$ by adding and deleting at most $\varepsilon n^2$ edges. It is still a challenging problem to determine the exact spectral extremal values of many non-bipartite graphs. Given a graph $F$ and an integer $p\geq 2$, the edge blow-up of $F$, denoted by $F^{p+1}$, is the graph obtained from replacing each edge in $F$ by a $K_{p+1}$ where the new vertices of $K_{p+1}$ are all distinct. In this paper, we determine the exact spectral extremal values of the edge blow-up of all non-bipartite graphs and provide the asymptotic spectral extremal values of the edge blow-up of all bipartite graphs for sufficiently large $n$, which can be seen as a spectral version of the theorem on ${\rm ex}(n,F^{p+1})$ given by Yuan [J. Combin. Theory Ser. B 152 (2022) 379--398]. As applications, on the one hand, we generalize several previous results on ${\rm spex}(n,F^{p+1})$ for $F$ being a matching and a star for $p\geq 3$. On the other hand, we obtain the exact values of ${\rm spex}(n,F^{p+1})$ for $F$ being a path, a cycle and a complete graph.

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arXiv2023

Spectral extremal problem on the square of $\ell$-cycle

Let $C_{\ell}$ be the cycle of order ${\ell}$. The square of $C_{\ell}$, denoted by $C_{\ell}^2$, is obtained by joining all pairs of vertices with distance no more than two in $C_{\ell}$. A graph is called $F$-free if it does not contain $F$ as a subgraph. Denote by $ex(n,F)$ and $spex(n,F)$ the maximum size and spectral radius over all $n$-vertex $F$-free graphs, respectively. The well-known Tur\'{a}n problem asks for the $ex(n,F)$, and Nikiforov in 2010 proposed a spectral counterpart, known as Brualdi-Solheid-Tur\'{a}n type problem, focusing on determining $spex(n,F)$. In this paper, we consider a Tur\'{a}n problem on $ex(n,C_{\ell}^2)$ and a Brualdi--Solheid--Tur\'{a}n type problem on $spex(n,C_{\ell}^2)$. We give a sharp bound of $ex(n,C_{\ell}^2)$ and $spex(n,C_{\ell}^2)$ for sufficiently large $n$, respectively. Moreover, in both results, we characterize the corresponding extremal graphs for any integer $\ell\geq 6$ that is not divisible by $3$.

math.CO↗

arXiv2023

Extremal spectral results of planar graphs without vertex-disjoint cycles

Given a planar graph family $\mathcal{F}$, let ${\rm ex}_{\mathcal{P}}(n,\mathcal{F})$ and ${\rm spex}_{\mathcal{P}}(n,\mathcal{F})$ be the maximum size and maximum spectral radius over all $n$-vertex $\mathcal{F}$-free planar graphs, respectively. Let $tC_{\ell}$ be the disjoint union of $t$ copies of $\ell$-cycles, and $t\mathcal{C}$ be the family of $t$ vertex-disjoint cycles without length restriction. Tait and Tobin [Three conjectures in extremal spectral graph theory, J. Combin. Theory Ser. B 126 (2017) 137--161] determined that $K_2+P_{n-2}$ is the extremal spectral graph among all planar graphs with sufficiently large order $n$, which implies the extremal graphs of both ${\rm spex}_{\mathcal{P}}(n,tC_{\ell})$ and ${\rm spex}_{\mathcal{P}}(n,t\mathcal{C})$ for $t\geq 3$ are $K_2+P_{n-2}$. In this paper, we first determine ${\rm spex}_{\mathcal{P}}(n,tC_{\ell})$ and ${\rm spex}_{\mathcal{P}}(n,t\mathcal{C})$ and characterize the unique extremal graph for $1\leq t\leq 2$, $\ell\geq 3$ and sufficiently large $n$. Secondly, we obtain the exact values of ${\rm ex}_{\mathcal{P}}(n,2C_4)$ and ${\rm ex}_{\mathcal{P}}(n,2\mathcal{C})$, which solve a conjecture of Li [Planar Tur\'an number of the disjoint union of cycles, Discrete Appl. Math. 342 (2024) 260--274] for $n\geq 2661$.

math.CO↗

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 Erd\H{o}s [Arch. Math. 13 (1962) 222--227] as well as the research on $ex(n,C_{2\ell+1})$ of F\"{u}redi and Gunderson [Combin. Probab. Comput. 24 (2015) 641--645]. On the other hand, we focus on the spectral Tur\'{a}n-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 Cioab\u{a}, Desai and Tait (2022).

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