arXiv subjects

Bo Ning

Publications and source records attributed to Bo Ning.

At least 55 records · Page 3Linked to original sources

arXiv2024

Variants of spectral Turán theorems and eigenvectors of graphs

In 2002, Nikiforov proved that for an $n$-vertex graph $G$ with clique number $ω$ and edge number $m$, the spectral radius $λ(G)$ satisfies $λ(G) \leq \sqrt{2(1 - 1/ω) m}$, which confirmed a conjecture implicitly suggested by Edwards and Elphick. In this paper, we prove a local version of spectral Turán inequality, which states that $λ^2(G)\leq 2\sum_{e\in E(G)}\frac{c(e)-1}{c(e)}$, where $c(e)$ is the order of the largest clique containing the edge $e$ in $G$. We also characterize the extremal graphs. We prove that our theorem implies Nikiforov's theorem and give an example to show that the difference of Nikiforov's bound and ours is $Ω(\sqrt{m})$ for some cases. Additionally, we establish a spectral counterpart to Ore's problem (1962) which asks for the maximum size of an $n$-vertex graph such that its complement is connected and does not contain $F$ as a subgraph. Our result leads to a new spectral Turán inequality applicable to graphs with connected complements. Finally, we disprove a conjecture of Gregory, asserting that for a connected $n$-vertex graph $G$ with chromatic number $k\geq 2$ and an independent set $S$, we have \[ \sum_{v\in S} x_v^2 \leq \frac{1}{2} - \frac{k-2}{2\sqrt{(k-2)^2 + 4(k-1)(n-k+1)}}, \] where $x_v$ is the component of the Perron vector of $G$ with respect to the vertex $v$. A modified version of Gregory's conjecture is proposed.

math.CO↗

arXiv2024

Monitoring the edges of product networks using distances

Foucaud {\it et al.} recently introduced and initiated the study of a new graph-theoretic concept in the area of network monitoring. Let $G$ be a graph with vertex set $V(G)$, $M$ a subset of $V(G)$, and $e$ be an edge in $E(G)$, and let $P(M, e)$ be the set of pairs $(x,y)$ such that $d_G(x, y)\neq d_{G-e}(x, y)$ where $x\in M$ and $y\in V(G)$. $M$ is called a \emph{distance-edge-monitoring set} if every edge $e$ of $G$ is monitored by some vertex of $M$, that is, the set $P(M, e)$ is nonempty. The {\em distance-edge-monitoring number} of $G$, denoted by $\operatorname{dem}(G)$, is defined as the smallest size of distance-edge-monitoring sets of $G$. For two graphs $G,H$ of order $m,n$, respectively, in this paper we prove that $\max\{m\operatorname{dem}(H),n\operatorname{dem}(G)\} \leq\operatorname{dem}(G\,\Box \,H) \leq m\operatorname{dem}(H)+n\operatorname{dem}(G) -\operatorname{dem}(G)\operatorname{dem}(H)$, where $\Box$ is the Cartesian product operation. Moreover, we characterize the graphs attaining the upper and lower bounds and show their applications on some known networks. We also obtain the distance-edge-monitoring numbers of join, corona, cluster, and some specific networks.

cs.DM↗

arXiv2024

Graph operations and a unified method for kinds of Turán-type problems on paths, cycles and matchings

Let $G$ be a connected graph and $\mathcal{P}(G)$ a graph parameter. We say that $\mathcal{P}(G)$ is feasible if $\mathcal{P}(G)$ satisfies the following properties: (I) $\mathcal{P}(G)\leq \mathcal{P}(G_{uv})$, if $G_{uv}=G[u\to v]$ for any $u,v$, where $G_{uv}$ is the graph obtained by applying Kelmans operation from $u$ to $v$; (II) $\mathcal{P}(G) <\mathcal{P}(G+e)$ for any edge $e\notin E(G)$. Let $P_k$ be a path of order $k$, $\mathcal{C}_{\geq k}$ the set of all cycles of length at least $k$ and $M_{k+1}$ a matching containing $k+1$ independent edges. In this paper, we mainly prove the following three results: (i) Let $n\geq k\geq 5$ and let $t=\left\lfloor\frac{k-1}{2}\right\rfloor$. Let $G$ be a $2$-connected $n$-vertex $\mathcal{C}_{\geq k}$-free graph with the maximum $\mathcal{P}(G)$ where $\mathcal{P}(G)$ is feasible. Then, $G\in \mathcal{G}^1_{n,k}=\{W_{n,k,s}=K_{s}\vee ((n-k+s)K_1\cup K_{k-2s}): 2\leq s\leq t\}$. (ii) Let $n\geq k\geq 4$ and let $t=\left\lfloor\frac{k}{2}\right\rfloor-1$. Let $G$ be a connected $n$-vertex $P_{k}$-free graph with the maximum $\mathcal{P}(G)$ where $\mathcal{P}(G)$ is feasible. Then, $G\in \mathcal{G}^2_{n,k}=\{W_{n,k-1,s}=K_{s}\vee ((n-k+s+1)K_1\cup K_{k-2s-1}): 1\leq s\leq t\}.$ (iii) Let $G$ be a connected $n$-vertex $M_{k+1}$-free graph with the maximum $\mathcal{P}(G)$ where $\mathcal{P}(G)$ is feasible. Then, $G\cong K_n$ when $n=2k+1$ and $G\in \mathcal{G}^3_{n,k}=\{K_s\vee ((n-2k+s-1)K_1\cup K_{2k-2s+1}):1\leq s\leq k\}$ when $n\geq 2k+2$. Directly derived from these three main results, we obtain a series of applications in Turán-type problems, generalized Turán-type problems, powers of graph degrees in extremal graph theory, and problems related to spectral radius, and signless Laplacian spectral radius in spectral graph theory.

math.CO↗

arXiv2023

Stability in Bondy's theorem on paths and cycles

In this paper, we study the stability result of a well-known theorem of Bondy. We prove that for any 2-connected non-hamiltonian graph, if every vertex except for at most one vertex has degree at least $k$, then it contains a cycle of length at least $2k+2$ except for some special families of graphs. Our results imply several previous classical theorems including a deep and old result by Voss. We point out our result on stability in Bondy's theorem can directly imply a positive solution (in a slight stronger form) to the following problem: Is there a polynomial time algorithm to decide whether a 2-connected graph $G$ on $n$ vertices has a cycle of length at least $\min\{2δ(G)+2,n\}$. This problem originally motivates the recent study on algorithmic aspects of Dirac's theorem by Fomin, Golovach, Sagunov and Simonov, although a stronger problem was solved by them by completely different methods. Our theorem can also help us to determine all extremal graphs for wheels on odd number of vertices. We also discuss the relationship between our results and some previous problems and theorems in spectral graph theory and generalized Turán problem.

math.CO↗

arXiv2023

Unsolved Problems in Spectral Graph Theory

Spectral graph theory is a captivating area of graph theory that employs the eigenvalues and eigenvectors of matrices associated with graphs to study them. In this paper, we present a collection of $20$ topics in spectral graph theory, covering a range of open problems and conjectures. Our focus is primarily on the adjacency matrix of graphs, and for each topic, we provide a brief historical overview.

math.CO↗

arXiv2023

Spectral Turán-type problems on sparse spanning graphs

Let $F$ be a graph and $\SPEX (n, F)$ be the class of $n$-vertex graphs which attain the maximum spectral radius and contain no $F$ as a subgraph. Let $\EX (n, F)$ be the family of $n$-vertex graphs which contain maximum number of edges and no $F$ as a subgraph. It is a fundamental problem in spectral extremal graph theory to characterize all graphs $F$ such that $\SPEX (n, F)\subseteq \EX (n, F)$ when $n$ is sufficiently large. Establishing the conjecture of Cioabă, Desai and Tait [European J. Combin., 2022], Wang, Kang, and Xue [J. Combin. Theory Ser. B, 2023] prove that: for any graph $F$ such that the graphs in $\EX (n, F)$ are Turán graphs plus $O(1)$ edges, $\SPEX (n, F)\subseteq \EX (n, F)$ for sufficiently large $n$. In this paper, we prove that $\SPEX (n, F)\subseteq \EX (n, F)$ for sufficiently large $n$, where $F$ is an $n$-vertex graph with no isolated vertices and $Δ(F) \leq \sqrt{n}/40$. We also prove a signless Laplacian spectral radius version of the above theorem. These results give new contribution to the open problem mentioned above, and can be seen as spectral analogs of a theorem of Alon and Yuster [J. Combin. Theory Ser. B, 2013]. Furthermore, as immediate corollaries, we have tight spectral conditions for the existence of several classes of special graphs, including clique-factors, $k$-th power of Hamilton cycles and $k$-factors in graphs. The first special class of graphs gives a positive answer to a problem of Feng, and the second one extends a previous result of Yan et al.

math.CO↗

arXiv2023

Weak Cosmic Censorship and Second Law of Black Hole Thermodynamics in Higher Derivative Gravity

Infalling matter may destroy a black hole and expose the naked singularity. Thus, Penrose proposed the weak cosmic censorship conjecture to avoid such a possibility. On the other hand, if the black hole is not destroyed by infalling matter, from the second law of black hole thermodynamics, the black hole entropy should increase due to the information carried by the infalling matter. In this work, we demonstrate by examples of perturbative near-extremal black holes in higher derivative gravity theories that the second law implies weak cosmic censorship. We also compare our proposal to the one developed by Sorce and Wald based on the first law of black hole thermodynamics and show that the latter fails to yield weak cosmic censorship in such cases. Finally, we give proof of our proposal for generic gravity theories.

hep-th↗

arXiv2023

Rainbow triangles sharing one common vertex or edge

Let $G$ be an edge-colored graph on $n$ vertices. For a vertex $v$, the \emph{color degree} of $v$ in $G$, denoted by $d^c(v)$, is the number of colors appearing on the edges incident with $v$. Denote by $δ^c(G)=\min\{d^c(v):v\in V(G)\}$. By a theorem of H. Li, an $n$-vertex edge-colored graph $G$ contains a rainbow triangle if $δ^c(G)\geq \frac{n+1}{2}$. Inspired by this result, we consider two related questions concerning edge-colored books and friendship subgraphs of edge-colored graphs. Let $k\geq 2$ be a positive integer. We prove that if $δ^c(G)\geq \frac{n+k-1}{2}$ where $n\geq 3k-2$, then $G$ contains $k$ rainbow triangles sharing one common edge; and if $δ^c(G)\geq \frac{n+2k-3}{2}$ where $n\geq 2k+9$, then $G$ contains $k$ rainbow triangles sharing one common vertex. The special case $k=2$ of both results improves H. Li's theorem. The main novelty of our proof of the first result is a combination of the recent new technique for finding rainbow cycles due to Czygrinow, Molla, Nagle, and Oursler and some recent counting technique from \cite{LNSZ}. The proof of the second result is with the aid of the machine implicitly in the work of Turán numbers for matching numbers due to Erdős and Gallai.

math.CO↗

arXiv2023

Eigenvalues and cycles of consecutive lengths

As the counterpart of classical theorems on cycles of consecutive lengths due to Bondy and Bollobás in spectral graph theory, Nikiforov proposed the following open problem in 2008: What is the maximum $C$ such that for all positive $\varepsilon \sqrt{\lfloor\frac{n^2}{4}\rfloor}$ contains a cycle of length $\ell$ for each integer $\ell\in[3,(C-\varepsilon)n]$. We prove that $C\geq\frac{1}{4}$ by a novel method, improving the existing bounds. Besides several novel ideas, our proof technique is partly inspirited by the recent research on Ramsey numbers of star versus large even cycles due to Allen, Łuczak, Polcyn and Zhang, and with aid of a powerful spectral inequality. We also derive an Erdős-Gallai-type edge number condition for even cycles, which may be of independent interest.

math.CO↗

arXiv2022

A revisit to Bang-Jensen-Gutin conjecture and Yeo's theorem

A path (cycle) is properly-colored if consecutive edges are of distinct colors. In 1997, Bang-Jensen and Gutin conjectured a necessary and sufficient condition for the existence of a Hamilton path in an edge-colored complete graph. This conjecture, confirmed by Feng, Giesen, Guo, Gutin, Jensen and Rafley in 2006, was laterly playing an important role in Lo's asymptotical proof of Bollobás-Erdős' conjecture on properly-colored Hamilton cycles. In 1997, Yeo obtained a structural characterization of edge-colored graphs that containing no properly colored cycles. This result is a fundamental tool in the study of edge-colored graphs. In this paper, we first give a much shorter proof of the Bang-Jensen-Gutin Conjecture by two novel absorbing lemmas. We also prove a new sufficient condition for the existence of a properly-colored cycle and then deduce Yeo's theorem from this result and a closure concept in edge-colored graphs.

math.CO↗

arXiv2022

DXM-TransFuse U-net: Dual Cross-Modal Transformer Fusion U-net for Automated Nerve Identification

Accurate nerve identification is critical during surgical procedures for preventing any damages to nerve tissues. Nerve injuries can lead to long-term detrimental effects for patients as well as financial overburdens. In this study, we develop a deep-learning network framework using the U-Net architecture with a Transformer block based fusion module at the bottleneck to identify nerve tissues from a multi-modal optical imaging system. By leveraging and extracting the feature maps of each modality independently and using each modalities information for cross-modal interactions, we aim to provide a solution that would further increase the effectiveness of the imaging systems for enabling the noninvasive intraoperative nerve identification.

eess.IV↗

arXiv2021

Counting substructures and eigenvalues II: quadrilaterals

Let $G$ be a graph and $λ(G)$ 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 $λ(G)>\sqrt{m}$. 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 $f(m)=Ω(m)$. In this article, by somewhat different techniques, we prove that $f(m)=Θ(m^2)$. 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.

math.CO↗

arXiv2021

Counting rainbow triangles in edge-colored graphs

Let $G$ be an edge-colored graph on $n$ vertices. The minimum color degree of $G$, denoted by $δ^c(G)$, is defined as the minimum number of colors assigned to the edges incident to a vertex in $G$. In 2013, H. Li proved that an edge-colored graph $G$ on $n$ vertices contains a rainbow triangle if $δ^c(G)\geq \frac{n+1}{2}$. In this paper, we obtain several estimates on the number of rainbow triangles through one given vertex in $G$. As consequences, we prove counting results for rainbow triangles in edge-colored graphs. One main theorem states that the number of rainbow triangles in $G$ is at least $\frac{1}{6}δ^c(G)(2δ^c(G)-n)n$, which is best possible by considering the rainbow $k$-partite Turán graph, where its order is divisible by $k$. This means that there are $Ω(n^2)$ rainbow triangles in $G$ if $δ^c(G)\geq \frac{n+1}{2}$, and $Ω(n^3)$ rainbow triangles in $G$ if $δ^c(G)\geq cn$ when $c>\frac{1}{2}$. Both results are tight in sense of the order of the magnitude. We also prove a counting version of a previous theorem on rainbow triangles under a color neighborhood union condition due to Broersma et al., and an asymptotically tight color degree condition forcing a colored friendship subgraph $F_k$ (i.e., $k$ rainbow triangles sharing a common vertex).

math.CO↗

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.

math.CO↗

arXiv2021

Extremal problems of Erdős, Faudree, Schelp and Simonovits on paths and cycles

For positive integers $n>d\geq k$, let $ϕ(n,d,k)$ denote the least integer $ϕ$ such that every $n$-vertex graph with at least $ϕ$ vertices of degree at least $d$ contains a path on $k+1$ vertices. Many years ago, Erdős, Faudree, Schelp and Simonovits proposed the study of the function $ϕ(n,d,k)$, and conjectured that for any positive integers $n>d\geq k$, it holds that $ϕ(n,d,k)\leq \lfloor\frac{k-1}{2}\rfloor\lfloor\frac{n}{d+1}\rfloor+ε$, where $ε=1$ if $k$ is odd and $ε=2$ otherwise. In this paper we determine the values of the function $ϕ(n,d,k)$ exactly. This confirms the above conjecture of Erdős et al. for all positive integers $k\neq 4$ and in a corrected form for the case $k=4$. Our proof utilizes, among others, a lemma of Erdős et al. \cite{EFSS89}, a theorem of Jackson \cite{J81}, and a (slight) extension of a very recent theorem of Kostochka, Luo and Zirlin \cite{KLZ}, where the latter two results concern maximum cycles in bipartite graphs. Moreover, we construct examples to provide answers to two closely related questions raised by Erdős et al.

math.CO↗

arXiv2021

Quasi-local Energy in 3D Gravity with Torsion

We show that for generic stationary spacetime and specific Killing fields, Wald's approach for quasi-local energy could be generalized to the first order formalism straightforwardly without introducing the Lorentz-Lie derivative. Via this approach, we derive the general formula for the black hole entropy in three dimensional torsional Mielke-Baekler gravity, reproducing precisely the total energy, the angular momentum as well as the black hole entropy for the BTZ-like solutions.

hep-th↗

arXiv2021

Constraints on low-energy effective theories from weak cosmic censorship

We examine the weak cosmic censorship conjecture (WCCC) for the extremal charged black hole in possible generalizations of Einstein-Maxwell theory due to the higher order corrections, up to fourth-derivative terms. Our derivation is based on Wald's gedanken experiment to destroy an extremal black hole. We find that, provided the null energy condition for the falling matter, the WCCC is preserved for all possible generalizations. Thus, the WCCC cannot serve as a constraint to the higher order effective theories. We also show that up to first order variations of black hole mass and charge, WCCC is preserved for non-rotating extremal black holes in all $n$-dimensional diffeomorphism-covariant theories of gravity and $U(1)$ gauge field.

gr-qc↗

arXiv2021

The stability method, eigenvalues and cycles of consecutive lengths

Woodall proved that for a graph $G$ of order $n\geq 2k+3$ where $k\geq 0$ is an integer, if $e(G)\geq \binom{n-k-1}{2}+\binom{k+2}{2}+1$ then $G$ contains a $C_{\ell}$ for each $\ell\in [3,n-k]$. In this article, we prove a stability result of this theorem. As a byproduct, we give complete solutions to two problems in \cite{GN19}. Our second part is devoted to an open problem by Nikiforov: what is the maximum $C$ such that for all positive $\varepsilon \sqrt{\lfloor\frac{n^2}{4}\rfloor}$ contains a cycle of length $\ell$ for every $\ell\leq (C-\varepsilon)n$. We prove that $C\geq\frac{1}{4}$ by a method different from previous ones, improving the existing bounds. We also derive an Erdős-Gallai type edge number condition for even cycles, which may be of independent interest.

math.CO↗