arXiv subjects

Yongtao Li

Publications and source records attributed to Yongtao Li.

At least 37 records · Page 2Linked to original sources

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 Turán number. Significant works due to Erdős, Rademacher, Lovász 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 Bollobás, Nikiforov, Ning and Zhai on counting triangles via the spectral radius, we study in this paper the spectral supersaturation problem for the bowtie. Let $λ(G)$ 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 Turán 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 $δ>0$ such that if $n$ is sufficiently large, $2\le q \le δ\sqrt{n}$, and $G$ is an $n$-vertex graph with $λ(G)\ge λ(K_{\lceil \frac{n}{2}\rceil, \lfloor \frac{n}{2}\rfloor}^q)$, 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 Bollobás-Nikiforov, we study the spectral supersaturation problem for color-critical graphs. Let $T_{n,r}$ be the $r$-partite Turán graph, let $\mathcal{T}_{n,r,q}$ denote the family of graphs obtained from $T_{n,r}$ by adding $q$ edges, and let $λ(G)$ 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 $δ_F > 0$ such that for sufficiently large $n$ and all $1 \leq q \leq δ_F \sqrt{n}$, any $n$-vertex graph $G$ with $λ(G) \ge \min_{T \in \mathcal{T}_{n,r,q}} λ(T)$ 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 $λ(G) \ge \max_{T \in \mathcal{T}_{n,r,q}} λ(T)$, valid for all $1 \leq q \leq δ_F n$. 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

Strong spectral stabilities for $C_{2k+1}$-free graphs

A stability result due to Ren, Wang, Wang and Yang [SIAM J. Discrete Math. 38 (2024)] shows that if $3\le r \le 2k$ and $n\ge 318 (r-2)^2k$, and $G$ is a $C_{2k+1}$-free graph on $n$ vertices with $e(G)\ge \lfloor {(n-r+1)^2}/{4}\rfloor +{r \choose 2}$, then $G$ can be made bipartite by deleting at most $r-2$ vertices. Using a different method, we give a linear bound on $n$ in terms of $k$ and show a stronger structural result, which roughly says that $G$ can be obtained from a large bipartite graph by suspending some small graphs that the total number of vertices is at most $r-2$. This improves a result of Yan and Peng (2024) by weakening the requirement on $n$ and $k$. As a direct corollary, we obtain a tight upper bound on the size of an $n$-vertex $C_{2k+1}$-free graph with chromatic number $χ(G)\ge r$ for every $r\le 2k$. The second part of this paper concerns the spectral extremal problem for $C_{2k+1}$-free graphs. We denote by $λ(G)$ the spectral radius of the adjacency matrix of a graph $G$. Let $T_{n-r+1,2}\circ K_r$ be the graph obtained by identifying a vertex of the complete graph $K_r$ and a vertex of the smaller partite set of the bipartite Turán graph $T_{n-r+1 ,2}$. Using the spectral techniques, we prove that if $3\le r\le 2k$ and $n\ge 712k$, and $G$ is an $n$-vertex $C_{2k+1}$-free graph with chromatic number $χ(G) \ge r$, then $λ(G)\le λ(T_{n-r+1,2}\circ K_r)$, where the equality holds if and only if $G=T_{n-r+1,2}\circ K_r$. Our result not only extends a result of Guo, Lin and Zhao [Linear Algebra Appl. 627 (2021)] as well as a result of Zhang and Zhao [Discrete Math. 346 (2023)], but also provides the first solution to the spectral extremal problem for $F$-free graphs with high chromatic number.

math.CO↗

arXiv2025

More on Nosal's spectral theorem: Books and $4$-cycles

Spectral graph theory studies how the eigenvalues of a graph relate to the structural properties of a graph. In this paper, we solve three open problems in spectral extremal graph theory which generalize the classical Turán-type supersaturation results. (a) We prove that every $m$-edge graph $G$ with the spectral radius $λ(G) > \sqrt{m}$ contains at least $\frac{1}{144} \sqrt{m}$ triangles sharing a common edge. This result confirms a conjecture of Nikiforov, and Li and Peng. Moreover, the bound is optimal up to a constant factor. (b) Next, for $m$-edge graph $G$ with $λ(G) > \sqrt{(1-\frac{1}{r})2m}$, we show that it must contain $Ω_r (\sqrt{m})$ copies of $K_{r+1}$ sharing $r$ common vertices. This confirms a conjecture of Li, Liu and Feng and unifies a series of spectral extremal results on books and cliques. Moreover, we also show that such a graph $G$ contains $Ω_r (m^{\frac{r-1}{2}})$ copies of $K_{r+1}$. This extends a result of Ning and Zhai for counting triangles. (c) We prove that every $m$-edge graph $G$ with $λ(G) > \sqrt{m}$ contains at least $(\frac{1}{8}-o(1)) m^2$ copies of 4-cycles, and we provide two constructions showing that the constant $\frac{1}{8}$ is the best possible. This result settles a problem raised by Ning and Zhai, and it gives the first asymptotics for counting degenerate bipartite graphs. The key to our proof are two structural results we obtain for graphs with large spectral radii on their maximum degree and on existence of large structured subgraphs, which we believe to be of independent interest.

math.CO↗

arXiv2025

A spectral Erdős-Faudree-Rousseau theorem

A well-known theorem of Mantel states that every $n$-vertex graph with more than $\lfloor n^2/4\rfloor $ edges contains a triangle. An interesting problem in extremal graph theory studies the minimum number of edges contained in triangles among graphs with a prescribed number of vertices and edges. Erdős, Faudree and Rousseau (1992) showed that a graph on $n$ vertices with more than $\lfloor n^2/4\rfloor $ edges contains at least $2\lfloor n/2\rfloor +1$ edges in triangles. Such edges are called triangular edges. In this paper, we present a spectral version of the result of Erdős, Faudree and Rousseau. Using the supersaturation-stability and the spectral technique, we prove that every $n$-vertex graph $G$ with $λ(G) \ge \sqrt{\lfloor n^2/4\rfloor}$ contains at least $2 \lfloor {n}/{2} \rfloor -1$ triangular edges, unless $G$ is a balanced complete bipartite graph. The method in our paper has some interesting applications. Firstly, the supersaturation-stability can be used to revisit a conjecture of Erdős concerning with the booksize of a graph, which was initially proved by Edwards (unpublished), and independently by Khadžiivanov and Nikiforov (1979). Secondly, our method can improve the bound on the order $n$ of the spectral extremal graph when we forbid the friendship graph as a substructure. We drop the condition that requires the order $n$ to be sufficiently large, which was investigated by Cioabă, Feng, Tait and Zhang (2020) using the triangle removal lemma. Thirdly, this method can be utilized to deduce the classical stability for odd cycles and it gives more concise bounds on parameters. Finally, the supersaturation-stability could be applied to deal with the spectral graph problems on counting triangles, which was recently studied by Ning and Zhai (2023).

math.CO↗

arXiv2025

Spectral supersaturation: Triangles and bowties

Recently, Ning and Zhai (2023) proved that every $n$-vertex graph $G$ with $λ(G) \ge \sqrt{\lfloor n^2/4\rfloor}$ has at least $\lfloor n/2\rfloor -1$ triangles, unless $G=K_{\lceil \frac{n}{2} \rceil, \lfloor \frac{n}{2} \rfloor}$. The aim of this paper is two-fold. Using the supersaturation-stability method, we prove a stability variant of Ning-Zhai's result by showing that such a graph $G$ contains at least $n-3$ triangles if no vertex is in all triangles of $G$. This result could also be viewed as a spectral version of a result of Xiao and Katona (2021). The second part concerns with the spectral supersaturation for the bowtie, which consists of two triangles sharing a common vertex. A theorem of Erdős, Füredi, Gould and Gunderson (1995) says that every $n$-vertex graph with more than $\lfloor n^2/4\rfloor +1$ edges contains a bowtie. For graphs of given order, the spectral supersaturation problem has not been considered for substructures that are not color-critical. In this paper, we give the first such theorem by counting the number of bowties. Let $K_{\lceil \frac{n}{2} \rceil, \lfloor \frac{n}{2} \rfloor}^{+2}$ be the graph obtained from $K_{\lceil \frac{n}{2} \rceil, \lfloor \frac{n}{2} \rfloor}$ by embedding two disjoint edges into the vertex part of size $\lceil \frac{n}{2} \rceil$. Our result shows that every graph $G$ with $n\ge 8.8 \times 10^6$ vertices and $λ(G)\ge λ(K_{\lceil \frac{n}{2} \rceil, \lfloor \frac{n}{2} \rfloor}^{+2})$ contains at least $\lfloor \frac{n}{2} \rfloor$ bowties, and $K_{\lceil \frac{n}{2} \rceil, \lfloor \frac{n}{2} \rfloor}^{+2}$ is the unique spectral extremal graph. This gives a spectral correspondence of a theorem of Kang, Makai and Pikhurko (2020). The method used in our paper provides a probable way to establish the spectral counting results for other graphs, even for non-color-critical graphs.

math.CO↗

arXiv2025

Spectral extremal problems for non-bipartite graphs without odd cycles

A well-known result of Mantel asserts that every $n$-vertex triangle-free graph $G$ has at most $\lfloor n^2/4 \rfloor$ edges. Moreover, Erdős proved that if $G$ is further non-bipartite, then $e(G)\le \lfloor {(n-1)^2}/{4}\rfloor +1$. Recently, Lin, Ning and Wu [Combin. Probab. Comput. 30 (2021)] established a spectral version by showing that if $G$ is a triangle-free non-bipartite graph on $n$ vertices, then $λ(G)\le λ(S_1(T_{n-1,2}))$, with equality if and only if $G=S_1(T_{n-1,2})$, where $S_1(T_{n-1,2})$ is obtained from $T_{n-1,2}$ by subdividing an edge. In this paper, we investigate the maximum spectral radius of a non-bipartite graph without some short odd cycles. Let $C_{2\ell +1}(T_{n-2\ell, 2})$ be the graph obtained by identifying a vertex of $C_{2\ell+1}$ and a vertex of the smaller partite set of $T_{n-2\ell ,2}$. We prove that for $1\le \ell < k$ and $n\ge 187k$, if $G$ is an $n$-vertex $\{C_3,\ldots ,C_{2\ell -1},C_{2k+1}\}$-free non-bipartite graph, then $λ(G)\le λ(C_{2\ell +1}(T_{n-2\ell, 2}))$, with equality if and only if $G=C_{2\ell +1}(T_{n-2\ell, 2})$. This result could be viewed as a spectral analogue of a min-degree result due to Yuan and Peng [European J. Combin. 127 (2025)]. Moreover, our result extends a result of Guo, Lin and Zhao [Linear Algebra Appl. 627 (2021)] as well as a recent result of Zhang and Zhao [Discrete Math. 346 (2023)] since we can get rid of the condition that $n$ is sufficiently large. The argument in our proof is quite different and makes use of the classical spectral stability method and the double-eigenvector technique. The main innovation lies in a more clever argument that guarantees a subgraph to be bipartite after removing few vertices, which may be of independent interest.

math.CO↗

arXiv2025

Maximal intersecting families revisited

The well-known Erdős--Ko--Rado theorem states that for $n> 2k$, every intersecting family of $k$-sets of $[n]:=\{1,\ldots ,n\}$ has at most $ {n-1 \choose k-1}$ sets, and the extremal family consists of all $k$-sets containing a fixed element (called a full star). The Hilton--Milner theorem provides a stability result by determining the maximum size of a uniform intersecting family that is not a subfamily of a full star. The further stabilities were studied by Han and Kohayakawa (2017) and Huang and Peng (2024). Two families $\mathcal{F}$ and $\mathcal{G}$ are called cross-intersecting if for every $F\in \mathcal{F}$ and $G\in \mathcal{G}$, the intersection $F\cap G$ is non-empty. Let $k \geq 1, t\ge 0$ and $n \geq 2 k+t$ be integers. Frankl (2016) proved that if $\mathcal{F} \subseteq\binom{[n]}{k+t}$ and $\mathcal{G} \subseteq\binom{[n]}{k}$ are cross-intersecting families, and $\mathcal{F}$ is non-empty and $(t+1)$-intersecting, then $|\mathcal{F}|+|\mathcal{G}| \leq\binom{n}{k}-\binom{n-k-t}{k}+1$. Recently, Wu (2023) sharpened Frankl's result by establishing a stability variant. The aim of this paper is two-fold. Inspired by the above results, we first prove a further stability variant that generalizes both Frankl's result and Wu's result. Secondly, as an interesting application, we illustrate that the aforementioned results on cross-intersecting families could be used to establish the stability results of the Erdős--Ko--Rado theorem. More precisely, we present new short proofs of the Hilton--Milner theorem, the Han--Kohayakawa theorem and the Huang--Peng theorem. Our arguments are more straightforward, and it may be of independent interest.

math.CO↗

arXiv2025

Stabilities of the Kleitman diameter theorem

Let $\mathcal{F}$ be a family of subsets of $[n]$. The diameter of $\mathcal{F}$ is the maximum size of symmetric differences among pairs of its members. Resolving a conjecture of Erdős, Kleitman determined the maximum size of a family with fixed diameter, which states that a family with diameter $s$ has cardinality at most that of a Hamming ball of radius $s/2$. Specifically, if $\mathcal{F} \subseteq 2^{[n]}$ is a family with diameter $s$, then for $s=2d$, $|\mathcal{F}|\le \sum_{i=0}^d {n \choose i}$; for $s=2d+1$, $|\mathcal{F}|\le \sum_{i=0}^d {n \choose i} + {n-1 \choose d}$. This result is known as the Kleitman diameter theorem, which generalizes both the Katona union theorem and the Erdős--Ko--Rado theorem. In 2017, Frankl provided a complete characterization of the extremal families of Kleitman's theorem and provided a stability result. In this paper, we determine the extremal families of Frankl's theorem and establish a further stability result of Kleitman's theorem. This solves a recent problem proposed by Li and Wu. Our findings constitute the second stability for the Kleitman diameter theorem.

math.CO↗

arXiv2025

A result for hemi-bundled cross-intersecting families

Two families $\mathcal{F}$ and $\mathcal{G}$ are called cross-intersecting if for every $F\in \mathcal{F}$ and $G\in \mathcal{G}$, the intersection $F\cap G$ is non-empty. It is significant to determine the maximum sum of sizes of cross-intersecting families under the additional assumption that one of the two families is intersecting. Such a pair of families is said to be hemi-bundled. In particular, Frankl (2016) proved that for $k \geq 1, t\ge 0$ and $n \geq 2 k+t$, if $\mathcal{F} \subseteq\binom{[n]}{k+t}$ and $\mathcal{G} \subseteq\binom{[n]}{k}$ are cross-intersecting families, in which $\mathcal{F}$ is non-empty and $(t+1)$-intersecting, then $|\mathcal{F}|+|\mathcal{G}| \leq\binom{n}{k}-\binom{n-k-t}{k}+1$. This bound can be attained when $\mathcal{F}$ consists of a single set. In this paper, we generalize this result under the constraint $|\mathcal{F}| \geq r$ for every $r\leq n-k-t+1$. Moreover, we investigate the stability results of Katona's theorem for non-uniform families with the $s$-union property. Our result extends the stabilities established by Frankl (2017) and Li and Wu (2024). As applications, we revisit a recent result of Frankl and Wang (2024) as well as a result of Kupavskii (2018). Furthermore, we determine the extremal families in these two results.

math.CO↗

arXiv2025

Proof of Frankl's conjecture on cross-intersecting families

Two families $\mathcal{F}$ and $\mathcal{G}$ are called cross-intersecting if for every $F\in \mathcal{F}$ and $G\in \mathcal{G}$, the intersection $F\cap G$ is non-empty. For any positive integers $n$ and $k$, let $\binom{[n]}{k}$ denote the family of all $k$-element subsets of $\{1,2,\ldots,n\}$. Let $t, s, k, n$ be non-negative integers with $k \geq s+1$ and $n \geq 2 k+t$. In 2016, Frankl proved that if $\mathcal{F} \subseteq\binom{[n]}{k+t}$ and $\mathcal{G} \subseteq\binom{[n]}{k}$ are cross-intersecting families, and $\mathcal{F}$ is $(t+1)$-intersecting and $|\mathcal{F}| \geq 1$, then $|\mathcal{F}|+|\mathcal{G}| \leq\binom{n}{k}-\binom{n-k-t}{k}+1$. Furthermore, Frankl conjectured that under an additional condition $\binom{[k+t+s]} {k+t}\subseteq\mathcal{F}$, the following inequality holds: $$ |\mathcal{F}|+|\mathcal{G}| \leq\binom{k+t+s}{k+t}+\binom{n}{k}-\sum_{i=0}^s\binom{k+t+s}{i}\binom{n-k-t-s}{k-i}. $$ In this paper, we prove this conjecture. The key ingredient is to establish a theorem for cross-intersecting families with a restricted universe. Moreover, we derive an analogous result for this conjecture.

math.CO↗

arXiv2025

The signless Laplacian spectral Turán problems for color-critical graphs

The well-known Turán theorem states that if $G$ is an $n$-vertex $K_{r+1}$-free graph, then $e(G)\le e(T_{n,r})$, with equality if and only if $G$ is the $r$-partite Turán graph $T_{n,r}$. A graph $F$ is called color-critical if it contains an edge whose deletion reduces its chromatic number. Extending the Turán theorem, Simonovits (1968) proved that for any color-critical graph $F$ with $χ(F)=r+1$ and sufficiently large $n$, the Turán graph $T_{n,r}$ is the unique graph with maximum number of edges among all $n$-vertex $F$-free graphs. Subsequently, Nikiforov [Electron. J. Combin., 16 (1) (2009)] proved a spectral version of the Simonovits theorem in terms of the adjacency spectral radius. In this paper, we show an extension of the Simonovits theorem for the signless Laplacian spectral radius. We prove that for any color-critical graph $F$ with $χ(F)=r+1\ge 4$ and sufficiently large $n$, if $G$ is an $F$-free graph on $n$ vertices, then $q(G)\le q(T_{n,r})$, with equality if and only if $G=T_{n,r}$. Our approach is to establish a signless Laplacian spectral version of the criterion of Keevash, Lenz and Mubayi [SIAM J. Discrete Math., 28 (4) (2014)]. Consequently, we can determine the signless Laplacian spectral extremal graphs for generalized books and even wheels. As an application, our result gives an upper bound on the degree power of an $F$-free graph. We show that if $n$ is sufficiently large and $G$ is an $F$-free graph on $n$ vertices with $m$ edges, then $\sum_{v\in V(G)} d^2(v) \le 2(1- \frac{1}{r})mn$, with equality if and only if $G$ is a regular Turán graph $T_{n,r}$. This extends a result of Nikiforov and Rousseau [J. Combin. Theory Ser B 92 (2004)].

math.CO↗

arXiv2024

Stabilities for non-uniform $t$-intersecting families

The study of intersection problems on families of sets is one of the most important topics in extremal combinatorics. As we all know, the extremal problems involving certain intersection constraints are equivalent to that with the union properties by taking complement of sets. A family of sets is called $s$-union if the union of any two sets in this family has size at most $s$. Katona [Acta Math. Hungar. 15 (1964)] provided the maximum size of an $s$-union family of sets of $[n]$, and he also determined the extremal families up to isomorphism. Recently, Frankl [J. Combin. Theory Ser. B 122 (2017) 869--876] sharpened this result by establishing the maximum size of an $s$-union family that is not a subfamily of the so-called Katona family. In this paper, we determine the maximum size of an $s$-union family that is neither contained in the Katona family nor in the Frankl family. Moreover, we characterize all extremal families achieving the upper bounds.

math.CO↗

arXiv2024

A spectral Erdős-Rademacher theorem

A classical result of Erdős and Rademacher (1955) indicates a supersaturation phenomenon. It says that if $G$ is a graph on $n$ vertices with at least $\lfloor {n^2}/{4} \rfloor +1$ edges, then $G$ contains at least $\lfloor {n}/{2}\rfloor$ triangles. We prove a spectral version of Erdős--Rademacher's theorem. Moreover, Mubayi [Adv. Math. 225 (2010)] extends the result of Erdős and Rademacher from a triangle to any color-critical graph. It is interesting to study the extension of Mubayi from a spectral perspective. However, it is not apparent to measure the increment on the spectral radius of a graph comparing to the traditional edge version (Mubayi's result). In this paper, we provide a way to measure the increment on the spectral radius of a graph and propose a spectral version on the counting problems for color-critical graphs.

math.CO↗

arXiv2024

AB-Type Dual-Phase High-Entropy Alloys as Negative Electrode of Ni-MH Batteries: Impact of Interphases on Electrochemical Performance

High-entropy alloys (HEAs) and their corresponding high-entropy hydrides are new potential candidates for negative electrode materials of nickel-metal hydride (Ni-MH) batteries. This study investigates the cyclic electrochemical hydrogen storage performance of two AB-type HEAs (A: hydride-forming elements, B: non-hydride-forming elements) in Ni-MH batteries. TiV2ZrCrMnFeNi with a dual-phase structure shows a fast activation and a low charge transfer impedance with a discharge capacity of 150 mAhg-1, while TiV1.5Zr1.5CrMnFeNi with a single phase shows a slow activation and a capacity of only 60 mAhg-1. The better electrochemical performance of TiV2ZrCrMnFeNi was attributed to its higher vanadium/zirconium ratio and abundant interphase boundaries, which act as hydrogen paths and heterogeneous hydride nucleation sites. These results suggest the high potential of dual-phase HEAs as new active electrode materials for Ni-MH batteries.

cond-mat.mtrl-sci↗

arXiv2024

$\mathcal{L}$-intersecting or Configuration Forbidden Families on Set Systems and Vector Spaces over Finite Fields

In this paper, we derive a tight upper bound for the size of an intersecting $k$-Sperner family of subspaces of the $n$-dimensional vector space $\mathbb{F}_{q}^{n}$ over finite field $\mathbb{F}_{q}$ which gives a $q$-analogue of the Erdős' $k$-Sperner Theorem, and we then establish a general relationship between upper bounds for the sizes of families of subsets of $[n] = \{1, 2, \dots, n\}$ with property $P$ and upper bounds for the sizes of families of subspaces of $\mathbb{F}_{q}^{n}$ with property $P$, where $P$ is either $\mathcal{L}$-intersecting or forbidding certain configuration. Applying this relationship, we derive generalizations of the well known results about the famous Erdős matching conjecture and Erdős-Chvátal simplex conjecture to linear lattices. As a consequence, we disprove a related conjecture on families of subspaces of $\mathbb{F}_{q}^{n}$ by Ihringer [Europ. J. Combin., 94 (2021), 103306].

math.CO↗

arXiv2024

Spectral extremal graphs for fan graphs

A well-known result of Nosal states that a graph $G$ with $m$ edges and $λ(G) > \sqrt{m}$ contains a triangle. Nikiforov [Combin. Probab. Comput. 11 (2002)] extended this result to cliques by showing that if $λ(G) > \sqrt{2m(1-1/r)}$, then $G$ contains a copy of $K_{r+1}$. Let $C_k^+$ be the graph obtained from a cycle $C_k$ by adding an edge to two vertices with distance two, and let $F_k$ be the friendship graph consisting of $k$ triangles that share a common vertex. Recently, Zhai, Lin and Shu [European J. Combin. 95 (2021)], Sun, Li and Wei [Discrete Math. 346 (2023)], and Li, Lu and Peng [Discrete Math. 346 (2023)] proved that if $ m\ge 8$ and $λ(G) \ge \frac{1}{2} (1+\sqrt{4m-3})$, then $G$ contains a copy of $C_5,C_5^+$ and $F_2$, respectively, unless $G=K_2\vee \frac{m-1}{2}K_1$. In this paper, we give a unified extension by showing that such a graph contains a copy of $V_5$, where $V_5=K_1\vee P_4$ is the join of a vertex and a path on four vertices. Our result extends the aforementioned results since $C_5,C_5^+$ and $F_2$ are proper subgraphs of $V_5$. In addition, we prove that if $m\ge 33$ and $λ(G) \ge 1+ \sqrt{m-2}$, then $G$ contains a copy of $F_3$, unless $G=K_3\vee \frac{m-3}{3}K_1$. This confirms a conjecture on the friendship graph $F_k$ in the case $k=3$. Finally, we conclude some spectral extremal graph problems concerning the large fan graphs and wheel graphs.

math.CO↗

arXiv2024

A spectral extremal problem on non-bipartite triangle-free graphs

A theorem of Nosal and Nikiforov states that if $G$ is a triangle-free graph with $m$ edges, then $λ(G)\le \sqrt{m}$, where the equality holds if and only if $G$ is a complete bipartite graph. A well-known spectral conjecture of Bollobás and Nikiforov [J. Combin. Theory Ser. B 97 (2007)] asserts that if $G$ is a $K_{r+1}$-free graph with $m$ edges, then $λ_1^2(G) + λ_2^2(G) \le (1-\frac{1}{r})2m$. Recently, Lin, Ning and Wu [Combin. Probab. Comput. 30 (2021)] confirmed the conjecture in the case $r=2$. Using this base case, they proved further that $λ(G)\le \sqrt{m-1}$ for every non-bipartite triangle-free graph $G$, with equality if and only if $m=5$ and $G=C_5$. Moreover, Zhai and Shu [Discrete Math. 345 (2022)] presented an improvement by showing $λ(G) \le β(m)$, where $β(m)$ is the largest root of $Z(x):=x^3-x^2-(m-2)x+m-3$. The equality in Zhai--Shu's result holds only if $m$ is odd and $G$ is obtained from the complete bipartite graph $K_{2,\frac{m-1}{2}}$ by subdividing exactly one edge. Motivated by this observation, Zhai and Shu proposed a question to find a sharp bound when $m$ is even. We shall solve this question by using a different method and characterize three kinds of spectral extremal graphs over all triangle-free non-bipartite graphs with even size. Our proof technique is mainly based on applying Cauchy interlacing theorem of eigenvalues of a graph, and with the aid of a triangle counting lemma in terms of both eigenvalues and the size of a graph.

math.CO↗