Related papers: Quadratic Embedding Constants of Strongly Regular …
This thesis focuses on theoretical and algorithmic tools for determining the numbers of induced subgraphs in strongly regular graphs, SRGs, and on further applications of such numbers. We consider in more detail a restricted class of these…
We discuss the problem of embedding graphs in the plane with restrictions on the vertex mapping. In particular, we introduce a technique for drawing planar graphs with a fixed vertex mapping that bounds the number of times edges bend. An…
Let $w:[0,1]^2\rightarrow [0,1]$ be a symmetric function, and consider the random process $G(n,w)$, where vertices are chosen from $[0,1]$ uniformly at random, and $w$ governs the edge formation probability. Such a random graph is said to…
For any positive definite rational quadratic form $q$ of $n$ variables let $G(\mathbb{Q}^n, q)$ denote the graph with vertices $\mathbb{Q}^n$ and $x, y \in \mathbb{Q}^n$ connected iff $q(x - y) = 1$. This notion generalises standard…
We show that for a sufficiently simple surface $S$, a right-angled Artin group $A(\Gamma)$ embeds into $\Mod(S)$ if and only if $\Gamma$ embeds into the curve graph $\mC(S)$ as an induced subgraph. When $S$ is sufficiently complicated,…
We say that a graph G is $(k,\ell)$-stable if removing $k$ vertices from it reduces its independence number by at most $\ell$. We say that G is tight $(k,\ell)$-stable if it is $(k,\ell)$-stable and its independence number equals…
Let $Q^N_l\subset \bC\bP^{N+1}$ denote the standard real, nondegenerate hyperquadric of signature $l$ and $M\subset \bC^{n+1}$ a real, Levi nondegenerate hypersurface of the same signature $l$. We shall assume that there is a holomorphic…
In this paper, we introduce a method called graph fusion embedding, designed for multi-graph embedding with shared vertex sets. Under the framework of supervised learning, our method exhibits a remarkable and highly desirable synergistic…
The consistency of effective models with QCD is investigated through the use of the QCD sum rule. Taking the potential model for the heavy quark system, we apply the method to two phenomenologically successful parameter sets, and obtain the…
We introduce a concept of an embedding of a quadratic space in an associative algebra. The general properties of such embeddings are analyzed by linking it to the Clifford algebra. Conversely, there isa simple description of the standard…
The strong geodetic number, $\text{sg}(G),$ of a graph $G$ is the smallest number of vertices such that by fixing one geodesic between each pair of selected vertices, all vertices of the graph are covered. In this paper, the study of the…
We show that if G is a 4-critical graph embedded in a fixed surface $\Sigma$ so that every contractible cycle has length at least 5, then G can be expressed as $G=G'\cup G_1\cup G_2\cup ... \cup G_k$, where $|V(G')|$ and $k$ are bounded by…
We study pseudo-geometric strongly regular graphs whose second subconstituent with respect to a vertex is a cover of a strongly regular graph or a complete graph. By studying the structure of such graphs, we characterize all graphs…
We consider the problem of embedding a relation, represented as a directed graph, into Euclidean space. For three types of embeddings motivated by the recent literature on knowledge graphs, we obtain characterizations of which relations…
Given a simplicial complex $K$, we consider several notions of geometric complexity of embeddings of $K$ in a Euclidean space ${\mathbb R}^d$: thickness, distortion, and refinement complexity (the minimal number of simplices needed for a PL…
A graph embedded in the 3-sphere is called irreducible if it is non-splittable and for any 2-sphere embedded in the 3-sphere that intersects the graph at one point the graph is contained in one of the 3-balls bounded by the 2-sphere. We…
Spectral embedding of graphs uses the top k non-trivial eigenvectors of the random walk matrix to embed the graph into R^k. The primary use of this embedding has been for practical spectral clustering algorithms [SM00,NJW02]. Recently,…
A graph with convex quadratic stability number is a graph for which the stability number is determined by solving a convex quadratic program. Since the very beginning, where a convex quadratic programming upper bound on the stability number…
In this paper we give a lower bound for the least distortion embedding of a distance regular graph into Euclidean space. We use the lower bound for finding the least distortion for Hamming graphs, Johnson graphs, and all strongly regular…
A recent result of Chepoi, Estellon and Vaxes [DCG '07] states that any planar graph of diameter at most 2R can be covered by a constant number of balls of size R; put another way, there are a constant-sized subset of vertices within which…