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The largest matching root of a $k$-graph is the largest real root of its matching polynomial, which is equal to the maximum modulus of all the zeros of the matching polynomial. In this paper, we investigate the perturbation of the largest matching root of $k$-graphs. We determine all $k$-graphs whose largest matching root attains the maximum among all $k$-cacti and linear $k$-cacti with a given number of cycles and edges, where a $k$-cactus is a $k$-graph in which every two distinct cycles have at most one vertex in common. To achieve this, we prove that the celebrated shifting operation of $k$-graphs, introduced by Erdős, Ko and Rado, does not decrease the largest matching root. This result extends a classical result by Csikvári (Electron. J. Combin. {\bf 18} (2011) $\#$P182) stating that the Kelmans transformation does not decrease the largest matching root of graphs.

In this paper, we propose a class of elementary plane geometry problems closely related to the title of this paper. Here, a circle is the 1-dimensional curve bounding a disk. For any nonnegative integer, a circle is called $n$-enclosing if it contains exactly $n$ lattice points on the $xy$-plane in its interior. The main questions are when the largest $n$-enclosing circle exists and what the largest radius is. We study the small integer cases by hand and extend to all $n<1100$ with the aid of a computer. We find that frequently such a circle does not exist, e.g., when $n=5,6$. We then show a few general results on these circles including some regularities among their radii and an easy criterion to determine exactly when largest $n$-enclosing circles exist. Further, from numerical evidence, we conjecture that the set of integers whose largest enclosing circles exist is infinite, and so is its complementary in the set of nonnegative integers. Throughout this paper we present more mysteries/problems/conjectures than answers/solutions/theorems. In particular, we list many conjectures and some unsolved problems including possible higher dimensional generalizations at the end of the l

The paper introduce a new type of partitions where the largest part appears exactly once, and the remaining parts constitute a partition of that largest part. We derive the generating function associated with these partitions and subsequently explore several variations, providing the corresponding generating functions for each variant.

We interpret the ranks of the rational homotopy groups of a K3 surface as dimensions of representations for the largest sporadic simple Mathieu group. We then construct a vertex algebra equipped with an action by the largest Mathieu group, and use it to associate Jacobi forms to this interpretation, in a compatible way. Our results suggest a topological role for the sporadic simple Mathieu groups in the theory of K3 surfaces.

We introduce a new approach for the adaptation of the Maximal Internal Envelope method, extended to address the Largest Empty Sphere problem within unstructured 3D point clouds. We explore the identification of the Largest Empty Sphere by computing Convex Hull vertices and employing a Voidness Score based on Minimal Distance Scoring for optimal segment selection. The integration of Delaunay triangulation and Voronoi diagrams facilitates the initial identification of potential Largest Empty Sphere candidates. Our analysis reveals the method's efficacy and efficiency, often locating the Largest Empty Sphere in initial computational stages, suggesting a lower complexity than initially projected.

The famous tree packing conjecture of Gyárfás from 1976 says that any sequence of trees $T_1,\ldots,T_n$ such that $|T_i|=i$ for each $i\in [n]$ packs into the complete $n$-vertex graph $K_n$. Packing even just the largest trees in such a sequence has proven difficult, with Bollobás drawing attention to this in 1995 by conjecturing that, for each $k$, if $n$ is sufficiently large then the largest $k$ trees in any such sequence can be packed into $K_n$. This has only been shown for $k\leq 5$, by Żak, despite many partial results and much related work on the full tree packing conjecture. We prove Bollobás's conjecture, by showing that, moreover, a linear number of the largest trees can be packed in the tree packing conjecture.

Let $G$ be a connected graph with vertex set $V(G)$. The distance, $d_G(u,v)$, between vertices $u$ and $v$ in $G$ is defined as the length of a shortest path between $u$ and $v$ in $G$. The distance matrix of $G$ is the matrix $D(G)=(d_G(u,v))_{u,v\in V(G)}$. The second largest distance eigenvalue of $G$ is the second largest one in the spectrum of $D(G)$. We show that any connected graph with the second largest distance eigenvalue less than $\frac{-3+\sqrt{5}}{2}$ is chordal, and characterize those bicyclic graphs and split graphs with the second largest distance eigenvalue less than $-\frac{1}{2}$.

This short note studies the fluctuations of the largest eigenvalue of symmetric random matrices with correlated Gaussian entries having positive mean. Under the assumption that the covariance kernel is absolutely summable, it is proved that the largest eigenvalue, after centering, converges in distribution to normal with an explicitly defined mean and variance. This result generalizes known findings for Wigner matrices with independent entries.

A bond of a graph $G$ is an inclusion-wise minimal disconnecting set of $G$, i.e., bonds are cut-sets that determine cuts $[S,V\setminus S]$ of $G$ such that $G[S]$ and $G[V\setminus S]$ are both connected. Given $s,t\in V(G)$, an $st$-bond of $G$ is a bond whose removal disconnects $s$ and $t$. Contrasting with the large number of studies related to maximum cuts, there are very few results regarding the largest bond of general graphs. In this paper, we aim to reduce this gap on the complexity of computing the largest bond and the largest $st$-bond of a graph. Although cuts and bonds are similar, we remark that computing the largest bond of a graph tends to be harder than computing its maximum cut. We show that {\sc Largest Bond} remains NP-hard even for planar bipartite graphs, and it does not admit a constant-factor approximation algorithm, unless $P = NP$. We also show that {\sc Largest Bond} and {\sc Largest $st$-Bond} on graphs of clique-width $w$ cannot be solved in time $f(w)\times n^{o(w)}$ unless the Exponential Time Hypothesis fails, but they can be solved in time $f(w)\times n^{O(w)}$. In addition, we show that both problems are fixed-parameter tractable when parameteriz

The symmetry frame formalism is an effective tool for computing the symmetries of a Riemann-Cartan geometry and, in particular, in metric teleparallel geometries. In the case of non-vanishing torsion in a four dimensional Riemann-Cartan geometry, the Minkowski geometry is the only geometry admitting ten affine frame symmetries. Excluding this geometry, the maximal number of affine frame symmetries is seven. A natural question is to ask what four dimensional geometries admit a seven-dimensional group of affine frame symmetries. Such geometries are locally homogeneous and admit the largest isotropy group permitted, and hence are called maximally isotropic. Using the symmetry frame formalism to compute affine frame symmetries along with the additional structure of the torsion tensor, we employ the Cartan-Karlhede algorithm to determine all possible seven-dimensional symmetry groups for Riemann-Cartan geometries.

A \emph{palindrome} is a word that reads the same forwards and backwards. A \emph{block palindrome factorization} (or \emph{BP-factorization}) is a factorization of a word into blocks that becomes palindrome if each identical block is replaced by a distinct symbol. We call the number of blocks in a BP-factorization the \emph{width} of the BP-factorization. The \emph{largest BP-factorization} of a word $w$ is the BP-factorization of $w$ with the maximum width. We study words with certain BP-factorizations. First, we give a recurrence for the number of length-$n$ words with largest BP-factorization of width $t$. Second, we show that the expected width of the largest BP-factorization of a word tends to a constant. Third, we give some results on another extremal variation of BP-factorization, the \emph{smallest BP-factorization}. A \emph{border} of a word $w$ is a non-empty word that is both a proper prefix and suffix of $w$. Finally, we conclude by showing a connection between words with a unique border and words whose smallest and largest BP-factorizations coincide.

In this paper, we estimate the largest Lyapunov exponent for open billiards in the plane. We show that the largest Lyapunov exponent is differentiable with respect to a billiard deformation.

We identify largest ideals in Leavitt path algebras: the largest locally left/right artinian (which is the largest semisimple one), the largest locally left/right noetherian without minimal idempotents, the largest exchange, and the largest purely infinite. This last ideal is described as a direct sum of purely infinite simple pieces plus purely infinite non-simple and non-decomposable pieces. The invariance under ring isomorphisms of these ideals is also studied.

In this paper we investigate the top-$k$-selection problem, i.e. determine the largest, second largest, ..., and the $k$-th largest elements, in the dynamic data model. In this model the order of elements evolves dynamically over time. In each time step the algorithm can only probe the changes of data by comparing a pair of elements. Previously only two special cases were studied[2]: finding the largest element and the median; and sorting all elements. This paper systematically deals with $k\in [n]$ and solves the problem almost completely. Specifically, we identify a critical point $k^*$ such that the top-$k$-selection problem can be solved error-free with probability $1-o(1)$ if and only if $k=o(k^*)$. A lower bound of the error when $k=Ω(k^*)$ is also determined, which actually is tight under some condition. On the other hand, it is shown that the top-$k$-set problem, which means finding the largest $k$ elements without sorting them, can be solved error-free for all $k\in [n]$. Additionally, we extend the dynamic data model and show that most of these results still hold.

In this paper, we show that the largest Laplacian H-eigenvalue of a $k$-uniform nontrivial hypergraph is strictly larger than the maximum degree when $k$ is even. A tight lower bound for this eigenvalue is given. For a connected even-uniform hypergraph, this lower bound is achieved if and only if it is a hyperstar. However, when $k$ is odd, it happens that the largest Laplacian H-eigenvalue is equal to the maximum degree, which is a tight lower bound. On the other hand, tight upper and lower bounds for the largest signless Laplacian H-eigenvalue of a $k$-uniform connected hypergraph are given. For a connected $k$-uniform hypergraph, the upper (respectively lower) bound of the largest signless Laplacian H-eigenvalue is achieved if and only if it is a complete hypergraph (respectively a hyperstar). The largest Laplacian H-eigenvalue is always less than or equal to the largest signless Laplacian H-eigenvalue. When the hypergraph is connected, the equality holds here if and only if $k$ is even and the hypergraph is odd-bipartite.

One of the main challenges for feature representation in deep learning-based classification is the design of appropriate loss functions that exhibit strong discriminative power. The classical softmax loss does not explicitly encourage discriminative learning of features. A popular direction of research is to incorporate margins in well-established losses in order to enforce extra intra-class compactness and inter-class separability, which, however, were developed through heuristic means, as opposed to rigorous mathematical principles. In this work, we attempt to address this limitation by formulating the principled optimization objective as learning towards the largest margins. Specifically, we firstly define the class margin as the measure of inter-class separability, and the sample margin as the measure of intra-class compactness. Accordingly, to encourage discriminative representation of features, the loss function should promote the largest possible margins for both classes and samples. Furthermore, we derive a generalized margin softmax loss to draw general conclusions for the existing margin-based losses. Not only does this principled framework offer new perspectives to under

Recently Friedman proved Alon's conjecture for many families of d-regular graphs, namely that given any epsilon > 0 `most' graphs have their largest non-trivial eigenvalue at most 2 sqrt{d-1}+epsilon in absolute value; if the absolute value of the largest non-trivial eigenvalue is at most 2 sqrt{d-1} then the graph is said to be Ramanujan. These graphs have important applications in communication network theory, allowing the construction of superconcentrators and nonblocking networks, coding theory and cryptography. As many of these applications depend on the size of the largest non-trivial positive and negative eigenvalues, it is natural to investigate their distributions. We show these are well-modeled by the beta=1 Tracy-Widom distribution for several families. If the observed growth rates of the mean and standard deviation as a function of the number of vertices holds in the limit, then in the limit approximately 52% of d-regular graphs from bipartite families should be Ramanujan, and about 27% from non-bipartite families (assuming the largest positive and negative eigenvalues are independent).

In this paper we prove that Amdeberhan's conjecture on the largest size of $(t, t+1, t+2)$-core partitions is true. We also show that the number of $(t, t + 1, t + 2)$-core partitions with the largest size is $1$ or $2$ based on the parity of $t$. More generally, the largest size of $(t,t+1,..., t+p)$-core partitions and the number of such partitions with the largest size are determined.

This short note presents upper bounds of the expectations of the largest singular values/eigenvalues of various types of random tensors in the non-asymptotic sense. For a standard Gaussian tensor of size $n_1\times\cdots\times n_d$, it is shown that the expectation of its largest singular value is upper bounded by $\sqrt {n_1}+\cdots+\sqrt {n_d}$. For the expectation of the largest $\ell^d$-singular value, it is upper bounded by $2^{\frac{d-1}{2}}\prod_{j=1}^{d}n_j^{\frac{d-2}{2d}}\sum^d_{j=1}n_j^{\frac{1}{2}}$. We also derive the upper bounds of the expectations of the largest Z-/H-($\ell^d$)/M-/C-eigenvalues of symmetric, partially symmetric, and piezoelectric-type Gaussian tensors, which are respectively upper bounded by $d\sqrt n$, $d\cdot 2^{\frac{d-1}{2}}n^{\frac{d-1}{2}}$, $2\sqrt m+2\sqrt n$, and $3\sqrt n$.