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Yongtao Li

Publications and source records attributed to Yongtao Li.

At least 55 records · Page 3Linked to original sources
arXiv2024

Extremal graphs for the odd prism

The Turán number $\mathrm{ex}(n,H)$ of a graph $H$ is the maximum number of edges in an $n$-vertex graph which does not contain $H$ as a subgraph. The Turán number of regular polyhedrons was widely studied in a series of works due to Simonovits. In this paper, we shall present the exact Turán number of the prism $C_{2k+1}^{\square} $, which is defined as the Cartesian product of an odd cycle $C_{2k+1}$ and an edge $ K_2 $. Applying a deep theorem of Simonovits and a stability result of Yuan [European J. Combin. 104 (2022)], we shall determine the exact value of $\mathrm{ex}(n,C_{2k+1}^{\square})$ for every $k\ge 1$ and sufficiently large $n$, and we also characterize the extremal graphs. Moreover, in the case of $k=1$, motivated by a recent result of Xiao, Katona, Xiao and Zamora [Discrete Appl. Math. 307 (2022)], we will determine the exact value of $\mathrm{ex}(n,C_{3}^{\square} )$ for every $n$ instead of for sufficiently large $n$.

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arXiv2023

Spectral radius of graphs forbidden $C_7$ or $C_6^{\triangle}$

Let $C_k^{\triangle}$ be the graph obtained from a cycle $C_{k}$ by adding a new vertex connecting two adjacent vertices in $C_{k}$. In this note, we obtain the graph maximizing the spectral radius among all graphs with size $m$ and containing no subgraph isomorphic to $C_6^{\triangle}$. As a byproduct, we will show that if the spectral radius $λ(G)\ge1+\sqrt{m-2}$, then $G$ must contains all the cycles $C_i$ for $3\le i\le 7$ unless $G\cong K_3\nabla \left(\frac{m-3}{3}K_1\right)$.

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arXiv2023

Refinement on spectral Turán's theorem

A well-known result in extremal spectral graph theory, due to Nosal and Nikiforov, states that if $G$ is a triangle-free graph on $n$ vertices, then $λ(G) \le λ(K_{\lfloor \frac{n}{2}\rfloor, \lceil \frac{n}{2} \rceil })$, equality holds if and only if $G=K_{\lfloor \frac{n}{2}\rfloor, \lceil \frac{n}{2} \rceil }$. Nikiforov [Linear Algebra Appl. 427 (2007)] extended this result to $K_{r+1}$-free graphs for every integer $r\ge 2$. This is known as the spectral Turán theorem. Recently, Lin, Ning and Wu [Combin. Probab. Comput. 30 (2021)] proved a refinement on this result for non-bipartite triangle-free graphs. In this paper, we provide alternative proofs for the result of Nikiforov and the result of Lin, Ning and Wu. Our proof can allow us to extend the later result to non-$r$-partite $K_{r+1}$-free graphs. Our result refines the theorem of Nikiforov and it also can be viewed as a spectral version of a theorem of Brouwer.

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arXiv2023

Spectral extremal graphs for the bowtie

Let $F_k$ be the (friendship) graph obtained from $k$ triangles by sharing a common vertex. The $F_k$-free graphs of order $n$ which attain the maximal spectral radius was firstly characterized by Cioabă, Feng, Tait and Zhang [Electron. J. Combin. 27 (4) (2020)], and later uniquely determined by Zhai, Liu and Xue [Electron. J. Combin. 29 (3) (2022)] under the condition that $n$ is sufficiently large. In this paper, we get rid of the condition on $n$ being sufficiently large if $k=2$. The graph $F_2$ is also known as the bowtie. We show that the unique $n$-vertex $F_2$-free spectral extremal graph is the balanced complete bipartite graph adding an edge in the vertex part with smaller size if $n\ge 7$, and the condition $n\ge 7$ is tight. Our result is a spectral generalization of a theorem of Erdős, Füredi, Gould and Gunderson [J. Combin. Theory Ser. B 64 (1995)], which states that $\mathrm{ex}(n,F_2)=\left\lfloor {n^2}/{4} \right\rfloor +1$. Moreover, we study the spectral extremal problem for $F_k$-free graphs with given number of edges. In particular, we show that the unique $m$-edge $F_2$-free spectral extremal graph is the join of $K_2$ with an independent set of $\frac{m-1}{2}$ vertices if $m\ge 8$, and the condition $m\ge 8$ is tight.

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arXiv2023

Spectral extremal graphs without intersecting triangles as a minor

Let $F_s$ be the friendship graph obtained from $s$ triangles by sharing a common vertex. For fixed $s\ge 2$ and sufficiently large $n$, the $F_s$-free graphs of order $n$ which attain the maximal spectral radius was firstly characterized by Cioabă, Feng, Tait and Zhang [Electron. J. Combin. 27 (4) (2020)],and later uniquely determined by Zhai, Liu and Xue [Electron. J. Combin. 29 (3) (2022)]. Recently, the spectral extremal problems was widely studied for graphs containing no $H$ as a minor. For instance, Tait [J. Combin. Theory Ser. A 166 (2019)], Zhai and Lin [J. Combin. Theory Ser. B 157 (2022)] solved the case $H=K_r$ and $H=K_{s,t}$, respectively. Motivated by these results, we consider the spectral extremal problems in the case $H=F_s$. We shall prove that $K_s \vee I_{n-s}$ is the unique graph that attain the maximal spectral radius over all $n$-vertex $F_s$-minor-free graphs. Moreover, let $Q_t$ be the graph obtained from $t$ copies of the cycle of length $4$ by sharing a common vertex. We also determine the unique $Q_t$-minor-free graph attaining the maximal spectral radius. Namely, $K_t \vee M_{n-t}$, where $M_{n-t}$ is a graph obtained from an independent set of order $n-t$ by embedding a matching consisting of $\lfloor \frac{n-t}{2}\rfloor$ edges.

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arXiv2022

Improvements on some partial trace inequalities for positive semidefinite block matrices

We study matrix inequalities involving partial traces for positive semidefinite block matrices. First of all, we present a new method to prove a celebrated result of Choi [Linear Algebra Appl. 516 (2017)]. Our method also allows us to prove a generalization of another result of Choi [Linear Multilinear Algebra 66 (2018)]. Furthermore, we shall give an improvement on a recent result of Li, Liu and Huang [Operators and Matrices 15 (2021)]. In addition, we include with some majorization inequalities involving partial traces for two by two block matrices, and also provide inequalities related to the unitarily invariant norms as well as the singular values, which can be viewed as slight extensions of two results of Lin [Linear Algebra Appl. 459 (2014)] and [Electronic J. Linear Algebra 31 (2016)].

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arXiv2022

Sufficient spectral conditions for graphs being $k$-edge-Hamiltonian or $k$-Hamiltonian

A graph $G$ is $k$-edge-Hamiltonian if any collection of vertex-disjoint paths with at most $k$ edges altogether belong to a Hamiltonian cycle in $G$. A graph $G$ is $k$-Hamiltonian if for all $S\subseteq V(G)$ with $|S|\le k$, the subgraph induced by $V(G)\setminus S$ has a Hamiltonian cycle. These two concepts are classical extensions for the usual Hamiltonian graphs. In this paper, we present some spectral sufficient conditions for a graph to be $k$-edge-Hamiltonian and $k$-Hamiltonian in terms of the adjacency spectral radius as well as the signless Laplacian spectral radius. Our results could be viewed as slight extensions of the recent theorems proved by Li and Ning [Linear Multilinear Algebra 64 (2016)], Nikiforov [Czechoslovak Math. J. 66 (2016)] and Li, Liu and Peng [Linear Multilinear Algebra 66 (2018)]. Moreover, we shall prove a stability result for graphs being $k$-Hamiltonian, which could be regarded as a complement of two recent results of Füredi, Kostochka and Luo [Discrete Math. 340 (2017)] and [Discrete Math. 342 (2019)].

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arXiv2022

Inequalities for generalized matrix function and inner product

We present inequalities related to generalized matrix function for positive semidefinite block matrices. We introduce partial generalized matrix functions corresponding to partial traces, and then provide a unified extension of the recent inequalities due to Lin [Electron. J. Linear Algebra 27 (2014) 821-826], Zhang et al. [Linear Algebra Appl. 498 (2016) 99-105] and [Electron. J. Linear Algebra 27 (2014) 332-341] and a result of Choi [Linear Algebra Appl. 532 (2017) 1-7]. Moreover, we demonstrate the application of a positive semidefinite $3\times 3$ block matrix, which motivates us to give alternative proofs of Dragomir's inequality and Krein's inequality.

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arXiv2022

A survey on spectral conditions for some extremal graph problems

This survey is two-fold. We first report new progress on the spectral extremal results on the Turán type problems in graph theory. More precisely, we shall summarize the spectral Turán function in terms of the adjacency spectral radius and the signless Laplacian spectral radius for various graphs. For instance, the complete graphs, general graphs with chromatic number at least three, complete bipartite graphs, odd cycles, even cycles, color-critical graphs and intersecting triangles. The second goal is to conclude some recent results of the spectral conditions on some graphical properties. By a unified method, we present some sufficient conditions based on the adjacency spectral radius and the signless Laplacian spectral radius for a graph to be Hamiltonian, $k$-Hamiltonian, $k$-edge-Hamiltonian, traceable, $k$-path-coverable, $k$-connected, $k$-edge-connected, Hamilton-connected, perfect matching and $β$-deficient.

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arXiv2022

The maximum spectral radius of non-bipartite graphs forbidding short odd cycles

It is well-known that eigenvalues of graphs can be used to describe structural properties and parameters of graphs. A theorem of Nosal states that if $G$ is a triangle-free graph with $m$ edges, then $λ(G)\le \sqrt{m}$, equality holds if and only if $G$ is a complete bipartite graph. Recently, Lin, Ning and Wu [Combin. Probab. Comput. 30 (2021)] proved a generalization for non-bipartite triangle-free graphs. Moreover, Zhai and Shu [Discrete Math. 345 (2022)] presented a further improvement. In this paper, we present an alternative method for proving the improvement by Zhai and Shu. Furthermore, the method can allow us to give a refinement on the result of Zhai and Shu for non-bipartite graphs without short odd cycles.

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arXiv2022

The spectral radius of graphs with no intersecting odd cycles

Let $H_{s,t_1,\ldots ,t_k}$ be the graph with $s$ triangles and $k$ odd cycles of lengths $t_1,\ldots ,t_k\ge 5$ intersecting in exactly one common vertex. Recently, Hou, Qiu and Liu [Discrete Math. 341 (2018) 126--137], and Yuan [J. Graph Theory 89 (1) (2018) 26--39] determined independently the maximum number of edges in an $n$-vertex graph that does not contain $H_{s,t_1,\ldots ,t_k}$ as a subgraph. In this paper, we determine the graphs of order $n$ that attain the maximum spectral radius among all graphs containing no $H_{s,t_1,\ldots ,t_k}$ for $n$ large enough.

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arXiv2022

New proofs of stability theorems on spectral graph problems

Both the Simonovits stability theorem and the Nikiforov spectral stability theorem are powerful tools for solving exact values of Turán numbers in extremal graph theory. Recently, Füredi [J. Combin. Theory Ser. B 115 (2015)] provided a concise and contemporary proof of the Simonovits stability theorem. In this note, we present a unified treatment for some extremal graph problems, including short proofs of Nikiforov's spectral stability theorem and the clique stability theorem proved recently by Ma and Qiu [European J. Combin. 84 (2020)]. Moreover, some spectral extremal problems related to the $p$-spectral radius and signless Laplacian radius are also included.

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arXiv2022

Extensions of some matrix inequalities related to trace and partial traces

We first present a determinant inequality related to partial traces for positive semidefinite block matrices. Our result extends a result of Lin [Czech. Math. J. 66 (2016)] and improves a result of Kuai [Linear Multilinear Algebra 66 (2018)]. Moreover, we provide a unified treatment of a result of Ando [ILAS Conference (2014)] and a recent result of Li, Liu and Huang [Operators and Matrices 15 (2021)]. Furthermore, we also extend some determinant inequalities involving partial traces to a larger class of matrices whose numerical ranges are contained in a sector. In addition, some extensions on trace inequalities for positive semidefinite $2\times 2$ block matrices are also included.

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arXiv2021

Extensions of Brunn-Minkovski's inequality to multiple matrices

Yuan and Leng (2007) gave a generalization of Ky Fan's determinantal inequality, which is a celebrated refinement of the fundamental Brunn-Minkowski inequality $(\det (A+B))^{1/n} \ge (\det A)^{1/n} +(\det B)^{1/n}$, where $A$ and $B$ are positive semidefinite matrices. In this note, we first give an extension of Yuan-Leng's result to multiple positive definite matrices, and then we further extend the result to a larger class of matrices whose numerical ranges are contained in a sector. Our result improves a recent result of Liu [Linear Algebra Appl. 508 (2016) 206--213].

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arXiv2021

A new matrix inequality involving partial traces

Let $A$ be an $m\times m$ positive semidefinite block matrix with each block being $n$-square. We write $\mathrm{tr}_1$ and $\mathrm{tr}_2$ for the first and second partial trace, respectively. In this paper, we prove the following inequality \[ (\mathrm{tr} A)I_{mn} - (\mathrm{tr}_2 A) \otimes I_n \ge \pm \bigl( I_m\otimes (\mathrm{tr}_1 A) -A\bigr).\] This inequality is not only a generalization of Ando's result [ILAS Conference (2014)] and Lin [Canad. Math. Bull. 59 (2016) 585--591], but it also could be regarded as a complement of a recent result of Choi [Linear Multilinear Algebra 66 (2018) 1619--1625]. Additionally, some new partial traces inequalities for positive semidefinite block matrices are also included.

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arXiv2021

An Oppenheim type inequality for positive definite block matrices

We present an Oppenheim type determinantal inequality for positive definite block matrices. Recently, Lin [Linear Algebra Appl. 452 (2014) 1--6] proved a remarkable extension of Oppenheim type inequality for block matrices, which solved a conjecture of Günther and Klotz. There is a requirement that two matrices commute in Lin's result. The motivation of this paper is to obtain another natural and general extension of Oppenheim type inequality for block matrices to get rid of the requirement that two matrices commute.

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arXiv2021

Spectral extremal graphs for intersecting cliques

The $(k,r)$-fan is the graph consisting of $k$ copies of the complete graph $K_r$ which intersect in a single vertex, and is denoted by $F_{k,r}$. Erdős, Füredi, Gould and Gunderson [J. Combin. Theory Ser. B 64 (1995) 89--100] determined the maximum number of edges in an $n$-vertex graph that does not contain $F_{k,3}$ as a subgraph. Furthermore, Chen, Gould, Pfender and Wei [J. Combin. Theory Ser. B 89 (2003) 159--171] proved the analogous result on $F_{k,r}$ for the general case $r\ge 3$.In this paper, we show that for sufficiently large $n$, the graphs of order $n$ that contain no copy of $F_{k,r}$ and attain the maximum spectral radius are also edge-extremal. That is, such graphs must have $\mathrm{ex}(n, F_{k,r})$ edges.

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