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Luo, Tian and Wu [Discrete Math. 345 (4) (2022) 112788] conjectured that for any tree $T$ with bipartition $(X,Y)$, every $k$-connected bipartite graph $G$ with minimum degree at least $k+w$, where $w=\max\{|X|,|Y|\}$, contains a tree $T'\cong T$ such that $\kappa(G-V(T'))\geq k$. In the paper, we confirm the conjecture for the spider by a new method, where a spider is a tree with at most one vertex of degree at least three., Comment: 6 pages. arXiv admin note: substantial text overlap with arXiv:2212.04637

Mental disorders present challenges in diagnosis and treatment due to their complex and heterogeneous nature. Electroencephalogram (EEG) has shown promise as a potential biomarker for these disorders. However, existing methods for analyzing EEG signals have limitations in addressing heterogeneity and capturing complex brain activity patterns between regions. This paper proposes a novel random effects state-space model (RESSM) for analyzing large-scale multi-channel resting-state EEG signals, accounting for the heterogeneity of brain connectivities between groups and individual subjects. We incorporate multi-level random effects for temporal dynamical and spatial mapping matrices and address nonstationarity so that the brain connectivity patterns can vary over time. The model is fitted under a Bayesian hierarchical model framework coupled with a Gibbs sampler. Compared to previous mixed-effects state-space models, we directly model high-dimensional random effects matrices without structural constraints and tackle the challenge of identifiability. Through extensive simulation studies, we demonstrate that our approach yields valid estimation and inference. We apply RESSM to a multi-site clinical trial of Major Depressive Disorder (MDD). Our analysis uncovers significant differences in resting-state brain temporal dynamics among MDD patients compared to healthy individuals. In addition, we show the subject-level EEG features derived from RESSM exhibit a superior predictive value for the heterogeneous treatment effect compared to the EEG frequency band power, suggesting the potential of EEG as a valuable biomarker for MDD.

The $k$-edge-colored bipartite Gallai-Ramsey number $\operatorname{bgr}_k(G:H)$ is defined as the minimum integer $n$ such that $n^2\geq k$ and for every $N\geq n$, every edge-coloring (using all $k$ colors) of complete bipartite graph $K_{N,N}$ contains a rainbow copy of $G$ or a monochromatic copy of $H$. In this paper, we first study the structural theorem on the complete bipartite graph $K_{n,n}$ with no rainbow copy of $K_{1,3}$. Next, we utilize the results to prove the exact values of $\operatorname{bgr}_{k}(P_4: H)$, $\operatorname{bgr}_{k}(P_5: H)$, $\operatorname{bgr}_{k}(K_{1,3}: H)$, where $H$ is a various union of cycles and paths and stars., Comment: 13 pages

Electroencephalogram (EEG) provides noninvasive measures of brain activity and is found to be valuable for diagnosis of some chronic disorders. Specifically, pre-treatment EEG signals in alpha and theta frequency bands have demonstrated some association with anti-depressant response, which is well-known to have low response rate. We aim to design an integrated pipeline that improves the response rate of major depressive disorder patients by developing an individualized treatment policy guided by the resting state pre-treatment EEG recordings and other treatment effects modifiers. We first design an innovative automatic site-specific EEG preprocessing pipeline to extract features that possess stronger signals compared with raw data. We then estimate the conditional average treatment effect using causal forests, and use a doubly robust technique to improve the efficiency in the estimation of the average treatment effect. We present evidence of heterogeneity in the treatment effect and the modifying power of EEG features as well as a significant average treatment effect, a result that cannot be obtained by conventional methods. Finally, we employ an efficient policy learning algorithm to learn an optimal depth-2 treatment assignment decision tree and compare its performance with Q-Learning and outcome-weighted learning via simulation studies and an application to a large multi-site, double-blind randomized controlled clinical trial, EMBARC.

Fujita and Kawarabayashi [J. Combin. Theory, Ser. B 98 (2008), 805--811] conjectured that for all positive integers $k$, $m$, there is a (least) non-negative integer $f_{k}(m)$ such that every $k$-connected graph $G$ with $\delta(G)\geq \lfloor\frac{3k}{2}\rfloor+ f_{k}(m)-1$ contains a connected subgraph $W$ of order $m$ such that $G-V(W)$ is still $k$-connected. Mader confirmed Fujita-Kawarabayashi's conjecture by proving $f_{k}(m)=m$ and $W$ is a path. In this paper, the authors will confirm Fujita-Kawarabayashi's conjecture again by proving $f_{k}(m)=m$ and $W$ is a spider by a new method, where a spider is a tree with at most one vertex of degree at least three. Meanwhile, this result will verify a conjecture proposed by Mader [J. Graph Theory 65 (2010), 61--69] for the case of the spider., Comment: 7 pages

A set of vertices $X\subseteq V$ in a simple graph $G(V,E)$ is irredundant (CO-irredundant) if each vertex $x\in X$ is either isolated in the induced subgraph $G[X]$ or else has a private neighbor $y\in V\setminus X$ ($y\in V$) that is adjacent to $x$ and to no other vertex of $X$. The irredundant Ramsey number $s(t_{1},\ldots,t_{l})$, CO-irredundant Ramsey number $s_{\operatorname{CO}}(t_{1},\ldots,t_{l})$, is the minimum $N$ such that every $l$-coloring of the edges of the complete graph $K_{N}$ on $N$ vertices has a monochromatic irredundant set, a monochromatic CO-irredundant set, of size $t_{i}$ for some $1\leq i\leq l$, respectively. In this paper, firstly, we establish a lower bound for the irredundant Ramsey number $s(t_{1},\ldots,t_{l})$ by a random and probabilistic method. Secondly, we improve an upper bound for $s(3,9)$ such that $24\leq s(3,9)\leq 26$. Thirdly, using Krivelevich's lemma, we establish an asymptotic lower bound for the $\operatorname{CO}$-irredundant Ramsey number $s_{\operatorname{CO}}(m,n)$., Comment: 19 pages,2 figures

An edge-colored graph $G$ is conflict-free connected if any two of its vertices are connected by a path, which contains a color used on exactly one of its edges. The conflict-free connection number of a connected graph $G$, denoted by $cfc(G)$, is defined as the minimum number of colors that are required in order to make $G$ conflict-free connected. In this paper, we investigate the relation between the conflict-free connection number and the independence number of a graph. We firstly show that $cfc(G)\le \alpha(G)$ for any connected graph $G$, and an example is given showing that the bound is sharp. With this result, we prove that if $T$ is a tree with $\Delta(T)\ge \frac{\alpha(T)+2}{2}$, then $cfc(T)=\Delta(T)$.

W. Mader [J. Graph Theory 65 (2010), 61--69] conjectured that for any tree $T$ of order $m$, every $k$-connected graph $G$ with $\delta(G)\geq\lfloor\frac{3k}{2}\rfloor+m-1$ contains a tree $T'\cong T$ such that $G-V(T')$ remains $k$-connected. In 2010, Mader confirmed the conjecture for the $k$-connected graph if $T$ is a path; very recently, Liu et al. confirmed the conjecture if $k=2,3$. The conjecture is open for $k\geq 4$ till now. In this paper, we show that Mader's conjecture is true for the $k+1$-connected graph if $T$ is a spider and $\Delta(G)=|G|-1$., Comment: 11 pages

Outlier detection is a fundamental task in data mining and has many applications including detecting errors in databases. While there has been extensive prior work on methods for outlier detection, modern datasets often have sizes that are beyond the ability of commonly used methods to process the data within a reasonable time. To overcome this issue, outlier detection methods can be trained over samples of the full-sized dataset. However, it is not clear how a model trained on a sample compares with one trained on the entire dataset. In this paper, we introduce the notion of resilience to sampling for outlier detection methods. Orthogonal to traditional performance metrics such as precision/recall, resilience represents the extent to which the outliers detected by a method applied to samples from a sampling scheme matches those when applied to the whole dataset. We propose a novel approach for estimating the resilience to sampling of both individual outlier methods and their ensembles. We performed an extensive experimental study on synthetic and real-world datasets where we study seven diverse and representative outlier detection methods, compare results obtained from samples versus those obtained from the whole datasets and evaluate the accuracy of our resilience estimates. We observed that the methods are not equally resilient to a given sampling scheme and it is often the case that careful joint selection of both the sampling scheme and the outlier detection method is necessary. It is our hope that the paper initiates research on designing outlier detection algorithms that are resilient to sampling., Comment: 18 pages