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Register a new userSigned graph embedding: when everybody can sit closer to friends than enemies
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Added by
Christopher Thraves Caro
Publication date
2014
fields
Informatics Engineering
and research's language is
English
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Signed graphs are graphs with signed edges. They are commonly used to represent positive and negative relationships in social networks. While balance theory and clusterizable graphs deal with signed graphs to represent social interactions, recent empirical studies have proved that they fail to reflect some current practices in real social networks. In this paper we address the issue of drawing signed graphs and capturing such social interactions. We relax the previous assumptions to define a drawing as a model in which every vertex has to be placed closer to its neighbors connected via a positive edge than its neighbors connected via a negative edge in the resulting space. Based on this definition, we address the problem of deciding whether a given signed graph has a drawing in a given $ell$-dimensional Euclidean space. We present forbidden patterns for signed graphs that admit the introduced definition of drawing in the Euclidean plane and line. We then focus on the $1$-dimensional case, where we provide a polynomial time algorithm that decides if a given complete signed graph has a drawing, and constructs it when applicable.
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We are given an integer $d$, a graph $G=(V,E)$, and a uniformly random embedding $f : V rightarrow {0,1}^d$ of the vertices. We are interested in the probability that $G$ can be realized by a scaled Euclidean norm on $mathbb{R}^d$, in the sense that there exists a non-negative scaling $w in mathbb{R}^d$ and a real threshold $theta > 0$ so that [ (u,v) in E qquad text{if and only if} qquad Vert f(u) - f(v) Vert_w^2 < theta,, ] where $| x |_w^2 = sum_i w_i x_i^2$. These constraints are similar to those found in the Euclidean minimum spanning tree (EMST) realization problem. A crucial difference is that the realization map is (partially) determined by the random variable $f$. In this paper, we consider embeddings $f : V rightarrow { x, y}^d$ for arbitrary $x, y in mathbb{R}$. We prove that arbitrary trees can be realized with high probability when $d = Omega(n log n)$. We prove an analogous result for graphs parametrized by the arboricity: specifically, we show that an arbitrary graph $G$ with arboricity $a$ can be realized with high probability when $d = Omega(n a^2 log n)$. Additionally, if $r$ is the minimum effective resistance of the edges, $G$ can be realized with high probability when $d=Omegaleft((n/r^2)log nright)$. Next, we show that it is necessary to have $d geq binom{n}{2}/6$ to realize random graphs, or $d geq n/2$ to realize random spanning trees of the complete graph. This is true even if we permit an arbitrary embedding $f : V rightarrow { x, y}^d$ for any $x, y in mathbb{R}$ or negative weights. Along the way, we prove a probabilistic analog of Radons theorem for convex sets in ${0,1}^d$. Our tree-realization result can complement existing results on statistical inference for gene expression data which involves realizing a tree, such as [GJP15].
The complexity of graph isomorphism (GraphIso) is a famous unresolved problem in theoretical computer science. For graphs $G$ and $H$, it asks whether they are the same up to a relabeling of vertices. In 1981, Lubiw proved that list restricted graph isomorphism (ListIso) is NP-complete: for each $u in V(G)$, we are given a list ${mathfrak L}(u) subseteq V(H)$ of possible images of $u$. After 35 years, we revive the study of this problem and consider which results for GraphIso translate to ListIso. We prove the following: 1) When GraphIso is GI-complete for a class of graphs, it translates into NP-completeness of ListIso. 2) Combinatorial algorithms for GraphIso translate into algorithms for ListIso: for trees, planar graphs, interval graphs, circle graphs, permutation graphs, bounded genus graphs, and bounded treewidth graphs. 3) Algorithms based on group theory do not translate: ListIso remains NP-complete for cubic colored graphs with sizes of color classes bounded by 8. Also, ListIso allows to classify results for the graph isomorphism problem. Some algorithms are robust and translate to ListIso. A fundamental problem is to construct a combinatorial polynomial-time algorithm for cubic graph isomorphism, avoiding group theory. By the 3rd result, ListIso is NP-hard for them, so no robust algorithm for cubic graph isomorphism exists, unless P = NP.
Electromagnetism (E&M) is often challenging for students enrolled in introductory college-level physics courses. Compared to mechanics, the mathematics of E&M is more sophisticated and the representations are more abstract. Furthermore, students may lack productive intuitions they had with force and motion. In this article, we explore the mathematization of electric charge. Specifically, we explore how difficulties with positive and negative signs can arise for learners who approach integers primarily as positions on a number line.
In 1998 the second author proved that there is an $epsilon>0$ such that every graph satisfies $chi leq lceil (1-epsilon)(Delta+1)+epsilonomegarceil$. The first author recently proved that any graph satisfying $omega > frac 23(Delta+1)$ contains a stable set intersecting every maximum clique. In this note we exploit the latter result to give a much shorter, simpler proof of the former. We include, as a certificate of simplicity, an appendix that proves all intermediate results with the exception of Halls Theorem, Brooks Theorem, the Lovasz Local Lemma, and Talagrands Inequality.
Using three supercomputers, we broke a record set in 2011, in the enumeration of non-isomorphic regular graphs by expanding the sequence of A006820 in Online Encyclopedia of Integer Sequences (OEIS), to achieve the number for 4-regular graphs of order 23 as 429,668,180,677,439, while discovering serval optimal regular graphs with minimum average shortest path lengths (ASPL) that can be used as interconnection networks for parallel computers. The number of 4-regular graphs and the optimal graphs, extremely time-consuming to calculate, result from a method we adapt from GENREG, a classical regular graph generator, to fit for supercomputers strengths of using thousands of processor cores.
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