Spectral Graph Theory
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Investigations on Unit Distance Property of Clebsch Graph and Its Complement
Proceedings of the World Congress on Engineering 2012 Vol I WCE 2012, July 4 - 6, 2012, London, U.K. Investigations on Unit Distance Property of Clebsch Graph and Its Complement Pratima Panigrahi and Uma kant Sahoo ∗yz Abstract|An n-dimensional unit distance graph is on parameters (16,5,0,2) and (16,10,6,6) respectively. a simple graph which can be drawn on n-dimensional These graphs are known to be unique in the respective n Euclidean space R so that its vertices are represented parameters [2]. In this paper we give unit distance n by distinct points in R and edges are represented by representation of Clebsch graph in the 3-dimensional Eu- closed line segments of unit length. In this paper we clidean space R3. Also we show that the complement of show that the Clebsch graph is 3-dimensional unit Clebsch graph is not a 3-dimensional unit distance graph. distance graph, but its complement is not. Keywords: unit distance graph, strongly regular The Petersen graph is the strongly regular graph on graphs, Clebsch graph, Petersen graph. parameters (10,3,0,1). This graph is also unique in its parameter set. It is known that Petersen graph is 2-dimensional unit distance graph (see [1],[3],[10]). It is In this article we consider only simple graphs, i.e. also known that Petersen graph is a subgraph of Clebsch undirected, loop free and with no multiple edges. The graph, see [[5], section 10.6]. study of dimension of graphs was initiated by Erdos et.al [3]. -
On the Tree–Depth of Random Graphs Arxiv:1104.2132V2 [Math.CO] 15 Feb 2012
On the tree–depth of random graphs ∗ G. Perarnau and O.Serra November 11, 2018 Abstract The tree–depth is a parameter introduced under several names as a measure of sparsity of a graph. We compute asymptotic values of the tree–depth of random graphs. For dense graphs, p n−1, the tree–depth of a random graph G is a.a.s. td(G) = n − O(pn=p). Random graphs with p = c=n, have a.a.s. linear tree–depth when c > 1, the tree–depth is Θ(log n) when c = 1 and Θ(log log n) for c < 1. The result for c > 1 is derived from the computation of tree–width and provides a more direct proof of a conjecture by Gao on the linearity of tree–width recently proved by Lee, Lee and Oum [?]. We also show that, for c = 1, every width parameter is a.a.s. constant, and that random regular graphs have linear tree–depth. 1 Introduction An elimination tree of a graph G is a rooted tree on the set of vertices such that there are no edges in G between vertices in different branches of the tree. The natural elimination scheme provided by this tree is used in many graph algorithmic problems where two non adjacent subsets of vertices can be managed independently. One good example is the Cholesky decomposition of symmetric matrices (see [?, ?, ?, ?]). Given an elimination tree, a distributed algorithm can be designed which takes care of disjoint subsets of vertices in different parallel processors. Starting arXiv:1104.2132v2 [math.CO] 15 Feb 2012 by the furthest level from the root, it proceeds by exposing at each step the vertices at a given depth. -
Three Conjectures in Extremal Spectral Graph Theory
Three conjectures in extremal spectral graph theory Michael Tait and Josh Tobin June 6, 2016 Abstract We prove three conjectures regarding the maximization of spectral invariants over certain families of graphs. Our most difficult result is that the join of P2 and Pn−2 is the unique graph of maximum spectral radius over all planar graphs. This was conjectured by Boots and Royle in 1991 and independently by Cao and Vince in 1993. Similarly, we prove a conjecture of Cvetkovi´cand Rowlinson from 1990 stating that the unique outerplanar graph of maximum spectral radius is the join of a vertex and Pn−1. Finally, we prove a conjecture of Aouchiche et al from 2008 stating that a pineapple graph is the unique connected graph maximizing the spectral radius minus the average degree. To prove our theorems, we use the leading eigenvector of a purported extremal graph to deduce structural properties about that graph. Using this setup, we give short proofs of several old results: Mantel's Theorem, Stanley's edge bound and extensions, the K}ovari-S´os-Tur´anTheorem applied to ex (n; K2;t), and a partial solution to an old problem of Erd}oson making a triangle-free graph bipartite. 1 Introduction Questions in extremal graph theory ask to maximize or minimize a graph invariant over a fixed family of graphs. Perhaps the most well-studied problems in this area are Tur´an-type problems, which ask to maximize the number of edges in a graph which does not contain fixed forbidden subgraphs. Over a century old, a quintessential example of this kind of result is Mantel's theorem, which states that Kdn=2e;bn=2c is the unique graph maximizing the number of edges over all triangle-free graphs. -
Maximizing the Order of a Regular Graph of Given Valency and Second Eigenvalue∗
SIAM J. DISCRETE MATH. c 2016 Society for Industrial and Applied Mathematics Vol. 30, No. 3, pp. 1509–1525 MAXIMIZING THE ORDER OF A REGULAR GRAPH OF GIVEN VALENCY AND SECOND EIGENVALUE∗ SEBASTIAN M. CIOABA˘ †,JACKH.KOOLEN‡, HIROSHI NOZAKI§, AND JASON R. VERMETTE¶ Abstract. From Alon√ and Boppana, and Serre, we know that for any given integer k ≥ 3 and real number λ<2 k − 1, there are only finitely many k-regular graphs whose second largest eigenvalue is at most λ. In this paper, we investigate the largest number of vertices of such graphs. Key words. second eigenvalue, regular graph, expander AMS subject classifications. 05C50, 05E99, 68R10, 90C05, 90C35 DOI. 10.1137/15M1030935 1. Introduction. For a k-regular graph G on n vertices, we denote by λ1(G)= k>λ2(G) ≥ ··· ≥ λn(G)=λmin(G) the eigenvalues of the adjacency matrix of G. For a general reference on the eigenvalues of graphs, see [8, 17]. The second eigenvalue of a regular graph is a parameter of interest in the study of graph connectivity and expanders (see [1, 8, 23], for example). In this paper, we investigate the maximum order v(k, λ) of a connected k-regular graph whose second largest eigenvalue is at most some given parameter λ. As a consequence of work of Alon and Boppana and of Serre√ [1, 11, 15, 23, 24, 27, 30, 34, 35, 40], we know that v(k, λ) is finite for λ<2 k − 1. The recent result of Marcus, Spielman, and Srivastava [28] showing the existence of infinite families of√ Ramanujan graphs of any degree at least 3 implies that v(k, λ) is infinite for λ ≥ 2 k − 1. -
Manifold Learning
MANIFOLD LEARNING Kavé Salamatian- LISTIC Learning Machine Learning: develop algorithms to automatically extract ”patterns” or ”regularities” from data (generalization) Typical tasks Clustering: Find groups of similar points Dimensionality reduction: Project points in a lower dimensional space while preserving structure Semi-supervised: Given labelled and unlabelled points, build a labelling function Supervised: Given labelled points, build a labelling function All these tasks are not well-defined Clustering Identify the two groups Dimensionality Reduction 4096 Objects in R ? But only 3 parameters: 2 angles and 1 for illumination. They span a 3- dimensional sub-manifold of R4096! Semi-Supervised Learning Assign labels to black points Supervised learning Build a function that predicts the label of all points in the space Formal definitions Clustering: Given build a function Dimensionality reduction: Given build a function Semi-supervised: Given and with m << n, build a function Supervised: Given , build Geometric learning Consider and exploit the geometric structure of the data Clustering: two ”close” points should be in the same cluster Dimensionality reduction: ”closeness” should be preserved Semi-supervised/supervised: two ”close” points should have the same label Examples: k-means clustering, k-nearest neighbors. Manifold ? A manifold is a topological space that is locally Euclidian Around every point, there is a neighborhood that is topologically isomorph to unit ball Regularization Adopt a functional viewpoint -
Contemporary Spectral Graph Theory
CONTEMPORARY SPECTRAL GRAPH THEORY GUEST EDITORS Maurizio Brunetti, University of Naples Federico II, Italy, [email protected] Raffaella Mulas, The Alan Turing Institute, UK, [email protected] Lucas Rusnak, Texas State University, USA, [email protected] Zoran Stanić, University of Belgrade, Serbia, [email protected] DESCRIPTION Spectral graph theory is a striking theory that studies the properties of a graph in relationship to the characteristic polynomial, eigenvalues and eigenvectors of matrices associated with the graph, such as the adjacency matrix, the Laplacian matrix, the signless Laplacian matrix or the distance matrix. This theory emerged in the 1950s and 1960s. Over the decades it has considered not only simple undirected graphs but also their generalizations such as multigraphs, directed graphs, weighted graphs or hypergraphs. In the recent years, spectra of signed graphs and gain graphs have attracted a great deal of attention. Besides graph theoretic research, another major source is the research in a wide branch of applications in chemistry, physics, biology, electrical engineering, social sciences, computer sciences, information theory and other disciplines. This thematic special issue is devoted to the publication of original contributions focusing on the spectra of graphs or their generalizations, along with the most recent applications. Contributions to the Special Issue may address (but are not limited) to the following aspects: f relations between spectra of graphs (or their generalizations) and their structural properties in the most general shape, f bounds for particular eigenvalues, f spectra of particular types of graph, f relations with block designs, f particular eigenvalues of graphs, f cospectrality of graphs, f graph products and their eigenvalues, f graphs with a comparatively small number of eigenvalues, f graphs with particular spectral properties, f applications, f characteristic polynomials. -
Testing Isomorphism in the Bounded-Degree Graph Model (Preliminary Version)
Electronic Colloquium on Computational Complexity, Revision 1 of Report No. 102 (2019) Testing Isomorphism in the Bounded-Degree Graph Model (preliminary version) Oded Goldreich∗ August 11, 2019 Abstract We consider two versions of the problem of testing graph isomorphism in the bounded-degree graph model: A version in which one graph is fixed, and a version in which the input consists of two graphs. We essentially determine the query complexity of these testing problems in the special case of n-vertex graphs with connected components of size at most poly(log n). This is done by showing that these problems are computationally equivalent (up to polylogarithmic factors) to corresponding problems regarding isomorphism between sequences (over a large alphabet). Ignoring the dependence on the proximity parameter, our main results are: 1. The query complexity of testing isomorphism to a fixed object (i.e., an n-vertex graph or an n-long sequence) is Θ(e n1=2). 2. The query complexity of testing isomorphism between two input objects is Θ(e n2=3). Testing isomorphism between two sequences is shown to be related to testing that two distributions are equivalent, and this relation yields reductions in three of the four relevant cases. Failing to reduce the problem of testing the equivalence of two distribution to the problem of testing isomorphism between two sequences, we adapt the proof of the lower bound on the complexity of the first problem to the second problem. This adaptation constitutes the main technical contribution of the current work. Determining the complexity of testing graph isomorphism (in the bounded-degree graph model), in the general case (i.e., for arbitrary bounded-degree graphs), is left open. -
Learning on Hypergraphs: Spectral Theory and Clustering
Learning on Hypergraphs: Spectral Theory and Clustering Pan Li, Olgica Milenkovic Coordinated Science Laboratory University of Illinois at Urbana-Champaign March 12, 2019 Learning on Graphs Graphs are indispensable mathematical data models capturing pairwise interactions: k-nn network social network publication network Important learning on graphs problems: clustering (community detection), semi-supervised/active learning, representation learning (graph embedding) etc. Beyond Pairwise Relations A graph models pairwise relations. Recent work has shown that high-order relations can be significantly more informative: Examples include: Understanding the organization of networks (Benson, Gleich and Leskovec'16) Determining the topological connectivity between data points (Zhou, Huang, Sch}olkopf'07). Graphs with high-order relations can be modeled as hypergraphs (formally defined later). Meta-graphs, meta-paths in heterogeneous information networks. Algorithmic methods for analyzing high-order relations and learning problems are still under development. Beyond Pairwise Relations Functional units in social and biological networks. High-order network motifs: Motif (Benson’16) Microfauna Pelagic fishes Crabs & Benthic fishes Macroinvertebrates Algorithmic methods for analyzing high-order relations and learning problems are still under development. Beyond Pairwise Relations Functional units in social and biological networks. Meta-graphs, meta-paths in heterogeneous information networks. (Zhou, Yu, Han'11) Beyond Pairwise Relations Functional units in social and biological networks. Meta-graphs, meta-paths in heterogeneous information networks. Algorithmic methods for analyzing high-order relations and learning problems are still under development. Review of Graph Clustering: Notation and Terminology Graph Clustering Task: Cluster the vertices that are \densely" connected by edges. Graph Partitioning and Conductance A (weighted) graph G = (V ; E; w): for e 2 E, we is the weight. -
Practical Verification of MSO Properties of Graphs of Bounded Clique-Width
Practical verification of MSO properties of graphs of bounded clique-width Ir`ene Durand (joint work with Bruno Courcelle) LaBRI, Universit´ede Bordeaux 20 Octobre 2010 D´ecompositions de graphes, th´eorie, algorithmes et logiques, 2010 Objectives Verify properties of graphs of bounded clique-width Properties ◮ connectedness, ◮ k-colorability, ◮ existence of cycles ◮ existence of paths ◮ bounds (cardinality, degree, . ) ◮ . How : using term automata Note that we consider finite graphs only 2/33 Graphs as relational structures For simplicity, we consider simple,loop-free undirected graphs Extensions are easy Every graph G can be identified with the relational structure (VG , edgG ) where VG is the set of vertices and edgG ⊆VG ×VG the binary symmetric relation that defines edges. v7 v6 v2 v8 v1 v3 v5 v4 VG = {v1, v2, v3, v4, v5, v6, v7, v8} edgG = {(v1, v2), (v1, v4), (v1, v5), (v1, v7), (v2, v3), (v2, v6), (v3, v4), (v4, v5), (v5, v8), (v6, v7), (v7, v8)} 3/33 Expression of graph properties First order logic (FO) : ◮ quantification on single vertices x, y . only ◮ too weak ; can only express ”local” properties ◮ k-colorability (k > 1) cannot be expressed Second order logic (SO) ◮ quantifications on relations of arbitrary arity ◮ SO can express most properties of interest in Graph Theory ◮ too complex (many problems are undecidable or do not have a polynomial solution). 4/33 Monadic second order logic (MSO) ◮ SO formulas that only use quantifications on unary relations (i.e., on sets). ◮ can express many useful graph properties like connectedness, k-colorability, planarity... Example : k-colorability Stable(X ) : ∀u, v(u ∈ X ∧ v ∈ X ⇒¬edg(u, v)) Partition(X1,..., Xm) : ∀x(x ∈ X1 ∨ . -
Harary Polynomials
numerative ombinatorics A pplications Enumerative Combinatorics and Applications ECA 1:2 (2021) Article #S2R13 ecajournal.haifa.ac.il Harary Polynomials Orli Herscovici∗, Johann A. Makowskyz and Vsevolod Rakitay ∗Department of Mathematics, University of Haifa, Israel Email: [email protected] zDepartment of Computer Science, Technion{Israel Institute of Technology, Haifa, Israel Email: [email protected] yDepartment of Mathematics, Technion{Israel Institute of Technology, Haifa, Israel Email: [email protected] Received: November 13, 2020, Accepted: February 7, 2021, Published: February 19, 2021 The authors: Released under the CC BY-ND license (International 4.0) Abstract: Given a graph property P, F. Harary introduced in 1985 P-colorings, graph colorings where each color class induces a graph in P. Let χP (G; k) counts the number of P-colorings of G with at most k colors. It turns out that χP (G; k) is a polynomial in Z[k] for each graph G. Graph polynomials of this form are called Harary polynomials. In this paper we investigate properties of Harary polynomials and compare them with properties of the classical chromatic polynomial χ(G; k). We show that the characteristic and the Laplacian polynomial, the matching, the independence and the domination polynomials are not Harary polynomials. We show that for various notions of sparse, non-trivial properties P, the polynomial χP (G; k) is, in contrast to χ(G; k), not a chromatic, and even not an edge elimination invariant. Finally, we study whether the Harary polynomials are definable in monadic second-order Logic. Keywords: Generalized colorings; Graph polynomials; Courcelle's Theorem 2020 Mathematics Subject Classification: 05; 05C30; 05C31 1. -
Interlacing Families I: Bipartite Ramanujan Graphs of All Degrees
Annals of Mathematics 182 (2015), 307{325 Interlacing families I: Bipartite Ramanujan graphs of all degrees By Adam W. Marcus, Daniel A. Spielman, and Nikhil Srivastava Abstract We prove that there exist infinite families of regular bipartite Ramanujan graphs of every degree bigger than 2. We do this by proving a variant of a conjecture of Bilu and Linial about the existence of good 2-lifts of every graph. We also establish the existence of infinite families of \irregular Ramanujan" graphs, whose eigenvalues are bounded by the spectral radius of their universal cover. Such families were conjectured to exist by Linial and others. In particular, we prove the existence of infinite families of (c; d)-biregular bipartite graphs with all nontrivial eigenvalues bounded by p p c − 1 + d − 1 for all c; d ≥ 3. Our proof exploits a new technique for controlling the eigenvalues of certain random matrices, which we call the \method of interlacing polynomials." 1. Introduction Ramanujan graphs have been the focus of substantial study in Theoret- ical Computer Science and Mathematics. They are graphs whose nontrivial adjacency matrix eigenvalues are as small as possible. Previous constructions of Ramanujan graphs have been sporadic, only producing infinite families of Ramanujan graphs of certain special degrees. In this paper, we prove a variant of a conjecture of Bilu and Linial [4] and use it to realize an approach they suggested for constructing bipartite Ramanujan graphs of every degree. Our main technical contribution is a novel existence argument. The con- jecture of Bilu and Linial requires us to prove that every graph has a signed adjacency matrix with all of its eigenvalues in a small range. -
Spectral Geometry for Structural Pattern Recognition
Spectral Geometry for Structural Pattern Recognition HEWAYDA EL GHAWALBY Ph.D. Thesis This thesis is submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy. Department of Computer Science United Kingdom May 2011 Abstract Graphs are used pervasively in computer science as representations of data with a network or relational structure, where the graph structure provides a flexible representation such that there is no fixed dimensionality for objects. However, the analysis of data in this form has proved an elusive problem; for instance, it suffers from the robustness to structural noise. One way to circumvent this problem is to embed the nodes of a graph in a vector space and to study the properties of the point distribution that results from the embedding. This is a problem that arises in a number of areas including manifold learning theory and graph-drawing. In this thesis, our first contribution is to investigate the heat kernel embed- ding as a route to computing geometric characterisations of graphs. The reason for turning to the heat kernel is that it encapsulates information concerning the distribution of path lengths and hence node affinities on the graph. The heat ker- nel of the graph is found by exponentiating the Laplacian eigensystem over time. The matrix of embedding co-ordinates for the nodes of the graph is obtained by performing a Young-Householder decomposition on the heat kernel. Once the embedding of its nodes is to hand we proceed to characterise a graph in a geometric manner. With the embeddings to hand, we establish a graph character- ization based on differential geometry by computing sets of curvatures associated ii Abstract iii with the graph nodes, edges and triangular faces.