Quadratic embedding constants of path graphs

图的二次嵌入常数：界限与距离谱

Quadratic embedding constants of graphs: Bounds and distance spectra

本文基于Schoenberg的经典工作研究了图的二次嵌入常数（QEC），给出了树的精确上下界，揭示了QEC与距离谱的关系，并推导了图簇的QEC计算方法。

关于二部图和theta图的二次可嵌入性

On quadratic embeddability of bipartite graphs and theta graphs

本文计算了去掉不相交边的完全二部图的二次嵌入常数，并研究了theta图的二次嵌入性质，最终构造出一类无限多的非二次可嵌入原始图。

通过二次嵌入常数与团图洞察的图分类

A Classification of Graphs through Quadratic Embedding Constants and Clique Graph Insights

本文利用二次嵌入常数和团图结构对图进行分类，证明 QEC 小于 -1/2 的图具有仙人掌状结构，并给出两极大团图的 QEC 公式及满足 QEC(G)<QEC(P_5) 的图刻画。

冠状图的二次嵌入常数

Quadratic Embedding Constants of Corona Graphs

本文研究了冠状图的二次嵌入常数，推导了其计算公式，并在特定条件下给出了显式表达式。

Determining Finite Connected Graphs Along the Quadratic Embedding Constants of Paths

The QE constant of a finite connected graph $G$, denoted by $\mathrm{QEC}(G)$, is by definition the maximum of the quadratic function associated to the distance matrix on a certain sphere of codimension two. We prove that the QE constants of paths $P_n$ form a strictly increasing sequence converging to $-1/2$. Then we formulate the problem of determining all the graphs $G$ satisfying $\mathrm{QEC}(P_n)\le\mathrm{QEC}(G)<\mathrm{QEC}(P_{n+1})$. The answer is given for $n=2$ and $n=3$ by exploiting forbidden subgraphs for $\mathrm{QEC}(G)<-1/2$ and the explicit QE constants of star products of t…

Embeddings of quadratic spaces over the field of $p$-adic numbers

Nondegenerate quadratic forms over $p$-adic fields are classified by their dimension, discriminant, and Hasse invariant. This paper uses these three invariants, elementary facts about $p$-adic fields and the theory of quadratic forms to determine which types of quadratic spaces -- including degenerate cases -- can be embedded in the Euclidean $p$-adic space $(\mathbb{Q}_{p}^{n},x_{1}^{2}+\cdots+x_{n}^{2})$, and the Lorentzian space $(\mathbb{Q}_{p}^{n},x_{1}^{2}+\cdots+x_{n-1}^{2}+λx_{n}^{2})$, where $\mathbb{Q}_{p}$ is the field of $p$-adic numbers, and $λ$ is a nonsquare in the finite field …

Embedding complete multi-partite graphs into Cartesian product of paths and cycles

Graph embedding is a powerful method in parallel computing that maps a guest network $G$ into a host network $H$. The performance of an embedding can be evaluated by certain parameters, such as the dilation, the edge congestion and the wirelength. In this manuscript, we obtain the wirelength (exact and minimum) of embedding complete multi-partite graphs into Cartesian product of paths and cycles, which include $n$-cube, $n$-dimensional mesh (grid), $n$-dimensional cylinder and $n$-dimensional torus, etc., as the subfamilies.

扇形图与图联的二次嵌入常数

Quadratic embedding constants of fan graphs and graph joins

本文推导了图联的二次嵌入常数一般公式，并应用于扇形图，通过新多项式的最小零点给出了其常数的精确表达式与奇偶性界。

多面体嵌入与其对偶结构之间的结构相似性及其在路径宽度自对偶性中的应用

Structural similarity between polyhedral embeddings and their duals and its application to self-duality of pathwidth

本文研究了多面体嵌入与其对偶图的结构相似性，证明了路径宽度之间的关系并改进了现有界。

Embedding hypercubes into torus and Cartesian product of paths and cycles for minimizing wirelength

Though embedding problems have been considered for several regular graphs, it is still an open problem for hypercube into torus. In the paper, we prove the conjecture mathematically and obtain the minimum wirelength of embedding for hypercube into Cartesian product of paths and/or cycles. In addition, we explain that Gray code embedding is an optimal strategy in such embedding problems.

有向图的构型空间与独立复形的拓扑持续性

Topological persistence of configuration spaces and independence complexes for digraphs

该研究探讨有向图及基础图的（路径）构型空间与独立复形的拓扑持续性，构建相关典范嵌入，将路径同调应用于路径独立复形，得到了关于Shannon容量的相关结论。

Unit-length embedding of cycles and paths on grid graphs

Although there are very algorithms for embedding graphs on unbounded grids, only few results on embedding or drawing graphs on restricted grids has been published. In this work, we consider the problem of embedding paths and cycles on grid graphs. We give the necessary and sufficient conditions for the existence of cycles of given length $k$ and paths of given length $k$ between two given vertices in $n$-vertex rectangular grid graphs and introduce two algorithms with running times O$(k)$ and O$(k^2)$ for finding respectively such cycles and paths. Also, we extend our results to $m\times n\tim…

Fault-Tolerant Path-Embedding of Twisted Hypercube-Like Networks THLNs

The twisted hypercube-like networks($THLNs$) contain several important hypercube variants. This paper is concerned with the fault-tolerant path-embedding of $n$-dimensional($n$-$D$) $THLNs$. Let $G_n$ be an $n$-$D$ $THLN$ and $F$ be a subset of $V(G_n)\cup E(G_n)$ with $|F|\leq n-2$. We show that for arbitrary two different correct vertices $u$ and $v$, there is a faultless path $P_{uv}$ of every length $l$ with $2^{n-1}-1\leq l\leq 2^n-f_v-1-α$, where $α=0$ if vertices $u$ and $v$ form a normal vertex-pair and $α=1$ if vertices $u$ and $v$ form a weak vertex-pair in $G_n-F$($n\geq5$).

关于嵌入的Kühnel猜想的二次估计

A quadratic estimation for the Kühnel conjecture on embeddings

本文针对Kühnel猜想的高维嵌入问题，在仅有线性估计的基础上，利用几何拓扑、组合学与线性代数的交叉作用，提出了$g\ge c_k n^2$的二次估计。

Shortest path embeddings of graphs on surfaces

The classical theorem of Fáry states that every planar graph can be represented by an embedding in which every edge is represented by a straight line segment. We consider generalizations of Fáry's theorem to surfaces equipped with Riemannian metrics. In this setting, we require that every edge is drawn as a shortest path between its two endpoints and we call an embedding with this property a shortest path embedding. The main question addressed in this paper is whether given a closed surface S, there exists a Riemannian metric for which every topologically embeddable graph admits a shortest pat…

Improved Lower Bounds for Graph Embedding Problems

In this paper, we give new, tight subexponential lower bounds for a number of graph embedding problems. We introduce two related combinatorial problems, which we call String Crafting and Orthogonal Vector crafting, and show that these cannot be solved in time $2^{o(|s|/\log{|s|})}$, unless the Exponential Time Hypothesis fails. These results are used to obtain simplified hardness results for several graph embedding problems, on more restricted graph classes than previously known: assuming the Exponential Time Hypothesis, there do not exist algorithms that run in $2^{o(n/\log n)}$ time for Subg…

Minimum Number of Bends of Paths of Trees in a Grid Embedding

We are interested in embedding trees T with maximum degree at most four in a rectangular grid, such that the vertices of T correspond to grid points, while edges of T correspond to non-intersecting straight segments of the grid lines. Such embeddings are called straight models. While each edge is represented by a straight segment, a path of T is represented in the model by the union of the segments corresponding to its edges, which may consist of a path in the model having several bends. The aim is to determine a straight model of a given tree T minimizing the maximum number of bends over all …

A quadratic identity in the shuffle algebra and an alternative proof for de Bruijn's formula

Motivated by a polynomial identity of certain iterated integrals, first observed in [CGM20] in the setting of lattice paths, we prove an intriguing combinatorial identity in the shuffle algebra. It has a close connection to de Bruijn's formula when interpreted in the framework of signatures of paths.