Riordan graphs II: Spectral properties

In a directed graph with the unique path property, there is a unique directed path of a given length between any ordered pair of its vertices. Such graphs, their paths, cycles, factors, and other properties, were extensively studied during the years, mainly in the context of de Bruijn graphs, forming the most important family of such graphs. Such a graph is equivalent to a binary matrix for which , where is the all‐ones matrix. In this paper, some new results on the structural properties of such graphs, which are not necessarily de Bruijn graphs, are proved. We mainly concentrate on such graphs for which each vertex has in‐degree two and out‐degree two. The minimum number and the maximum number of vertices which are reached from a given vertex using all paths of length smaller than are considered. We provide an exact formula for the number of factors in these graphs and show that this number depends only on the number of alternating cycles in the graph, where in an alternating cycle every two consecutive edges have opposite directions in the cycle. Constructions of such graphs using several methods are presented. In particular, we present a new construction which provides a large set of such graphs with long alternating cycles. We define the integral graph and the derivative graph and implement these definitions on this family of graphs. Various properties from the point of view of the de Bruijn graph and the theory of nonsingular shift‐registers sequences are also considered. Finally, generalizations for graphs in which the in‐degree and the out‐degree of each vertex are greater than two are also discussed.

From social networks to machine learning to biological network analysis, graphs spring up in all sorts of applications related to either the natural sciences or systems created by humans. While individual nodes and edges express relationships between interacting entities, there is often far greater insight in understanding a graph's larger structural properties, e.g., treewidths, the maximal degree, or the second smallest eigenvalue of an associated graph Laplacian. As these fundamental structural properties are independent of a specific node labeling, it is then worthwhile to examine graph invariants, functions of graphs that do not depend on specific node labels. The paper by Chandrasekaran, Parrilo, and Willsky explores the realm of convex graph invariants, convex functions of a graph's adjacency matrix which are also graph invariants. These convex graph invariants arise in many practical situations and are advantageous in that one can leverage convex optimization ideas and tools to provide tractable algorithms. Three canonical graph structural problems are introduced: graph deconvolution, graph generation, and graph hypothesis testing. Deconvolution seeks to unravel a graph into simpler graph structures. Network analysis in areas such as biological sciences and social networks often rely on this type of decomposition. Graph generation seeks to produce graphs satisfying specified constraints (e.g., each node has a certain degree or the corresponding Laplacian matrix has certain spectral properties). This might arise in situations where one wishes to construct an economical network with certain robustness characteristics. Graph hypothesis testing aims to categorize candidate graphs, i.e., deciding whether they fit better within one graph family or another (where families are specified by graph invariants). The paper argues that the key to understanding and computing convex graph invariants lies in being able to first represent them via elementary invariants. This leads to solution methodologies associated with quadratic assignment problems and tractable approximations via optimization techniques such as linear programming and semidefinite programming. Overall, the paper provides a clear view into the world of convex graph invariants. In addition to providing a general framework for computing these invariants, numerous examples introduce readers unfamiliar with this area to the wide range of practical cases where the notions of invariant convex sets and convex graph invariants emerge.

Let Gr,p be a graph obtained from a path by adjoining a cycle Cr of length r to one end and the central vertex of a star Sp on p vertices to the other end.In this paper, it is proven that unicyclic graph Gr,p with r even is determined by its Laplacian spectrum except for n = p + 4.

Structural properties of Toeplitz graphs

Consider a simple graph [Formula: see text] of order [Formula: see text] and size [Formula: see text] having Laplacian eigenvalues [Formula: see text]. Let [Formula: see text] be the sum of [Formula: see text] largest Laplacian eigenvalues of [Formula: see text]. Brouwer conjectured that [Formula: see text] for all [Formula: see text]. We obtain an upper bound for [Formula: see text] in terms of the clique number [Formula: see text], the number of vertices [Formula: see text] and the non-negative integers [Formula: see text] associated to the structure of the graph [Formula: see text]. We show that the Brouwer’s conjecture holds true for some new families of graphs. We use the same technique to prove that the Brouwer’s conjecture is true for a subclass of split graphs (It is already known that Brouwer’s conjecture holds for split graphs).

Traditionally, graph algorithms get a single graph as input, and then they should decide if this graph satisfies a certain property \(\varPhi \). What happens if this question is modified in a way that we get a possibly infinite family of graphs as an input, and the question is if there exists one graph satisfying \(\varPhi \)? We approach this question by using formal languages for specifying families of graphs. In particular, we show that certain graph properties can be decided by studying the syntactic monoid of the specification language.

Extremality of degree-based graph entropies

Distance-balanced graphs: Symmetry conditions

The Fibonacci dimension ${\rm fdim}(G)$ of a graph $G$ is introduced as the smallest integer $f$ such that $G$ admits an isometric embedding into $\Gamma_f$, the $f$-dimensional Fibonacci cube. We give bounds on the Fibonacci dimension of a graph in terms of the isometric and lattice dimension, provide a combinatorial characterization of the Fibonacci dimension using properties of an associated graph, and establish the Fibonacci dimension for certain families of graphs. From the algorithmic point of view, we prove that it is NP-complete to decide whether ${\rm fdim}(G)$ equals the isometric dimension of $G$, and show that no algorithm to approximate ${\rm fdim}(G)$ has approximation ratio below $741/740$, unless P$=$NP. We also give a $(3/2)$-approximation algorithm for ${\rm fdim}(G)$ in the general case and a $(1+\varepsilon)$-approximation algorithm for simplex graphs.

A natural way to describe a family of languages is to use rational transformations from a generator. From these transformations, Ginsburg and Greibach have defined the Abstract Family of Languages (AFL). Infinite graphs (also called infinite automata) are natural tools to study languages. In this paper, we study families of infinite graphs that are described from generators by transformations preserving the decidability of monadic second order logic. We define the Abstract Family of Graphs (AFG). We show that traces of AFG are rational cones and traces of AFG that admit a rationally colored generator are AFL. We generalize some properties of prefix recognizable graphs to AFG. We apply these tools and the notion of geometrical complexity to study subfamilies of prefix recognizable graphs.

Classification of subsets with minimal width and dual width in Grassmann, bilinear forms and dual polar graphs

Consider property testing on bounded degree graphs and let $\\varepsilon>0$\ndenote the proximity parameter. A remarkable theorem of Newman-Sohler (SICOMP\n2013) asserts that all properties of planar graphs (more generally hyperfinite)\nare testable with query complexity only depending on $\\varepsilon$. Recent\nadvances in testing minor-freeness have proven that all additive and monotone\nproperties of planar graphs can be tested in $poly(\\varepsilon^{-1})$ queries.\nSome properties falling outside this class, such as Hamiltonicity, also have a\nsimilar complexity for planar graphs. Motivated by these results, we ask: can\nall properties of planar graphs can be tested in $poly(\\varepsilon^{-1})$\nqueries? Is there a uniform query complexity upper bound for all planar\nproperties, and what is the "hardest" such property to test? We discover a\nsurprisingly clean and optimal answer. Any property of bounded degree planar\ngraphs can be tested in $\\exp(O(\\varepsilon^{-2}))$ queries. Moreover, there is\na matching lower bound, up to constant factors in the exponent. The natural\nproperty of testing isomorphism to a fixed graph needs\n$\\exp(\\Omega(\\varepsilon^{-2}))$ queries, thereby showing that (up to\npolynomial dependencies) isomorphism to an explicit fixed graph is the hardest\nproperty of planar graphs. The upper bound is a straightforward adapation of\nthe Newman-Sohler analysis that tracks dependencies on $\\varepsilon$ carefully.\nThe main technical contribution is the lower bound construction, which is\nachieved by a special family of planar graphs that are all mutually far from\neach other. We can also apply our techniques to get analogous results for\nbounded treewidth graphs. We prove that all properties of bounded treewidth\ngraphs can be tested in $\\exp(O(\\varepsilon^{-1}\\log \\varepsilon^{-1}))$\nqueries. Moreover, testing isomorphism to a fixed forest requires\n$\\exp(\\Omega(\\varepsilon^{-1}))$ queries.\n

In recent years, considerable research has been carried out relative to the electromagnetic (EM) propagation and refraction characteristics in metamaterials with emphasis on the origins of negative refractive index. Negative refractive index may be introduced in metamaterials via different methods; one such is the con- dition whereby the Poynting vector of the EM wave is in opposition to the group velocity in the material. Alternatively, negative refractive index also occurs when the group and phase velocities in the medium are in opposition. The latter phenomenon has been extensively investigated in the literature, including recent work involving chiral metamaterials with material dispersion up to the first order. This paper examines the pos- sible emergence of negative refractive index in dispersive chiral metamaterials with material dispersion up to the second order. The motivation is to determine if using second- as opposed to first-order dispersion may lead to more practical negative index behavior. A spectral approach combined with a slowly time-varying phasor analy- sis is applied, leading to the analytic derivation of EM phase and group velocities, and the resulting phase and group velocities and the corresponding phase and group indices are evaluated by selecting somewhat arbitrary dispersive parameters. The results indicate the emergence of negative index (via negative phase indices along with positive group indices, as reported in the literature) or negative index material (NIM) behavior over infor- mation bandwidths in the low RF range. The second-order results are not significantly better than those for first-order results based on the theoretical analysis; however, greater parametric flexibility exists for the second-order system leading to the higher likelihood of achieving NIM over practical frequency bands. The velocities and indices computed using the Lorentzian and Condon models yield an NIM bandwidth around 200 − 400 Mrad∕sec, about 2 orders of magnitude higher than that for the parametric approach; more impor- tantly, NIM is found not to occur in the first order when using practical models. © 2015 Society of Photo-Optical

Kneser graphs are like Swiss cheese, Discrete Analysis 2018:2, 18 pp. This paper relates two very interesting areas of research in extremal combinatorics: removal lemmas, and of variables. A _removal lemma_ is a result that states that if $S$ is a set that contains only few copies of a certain structure, then one can remove just a few elements from $S$ to create a set $S'$ that has no copies of that structure. For instance, the triangle removal states that a graph with few triangles can be approximated by a graph with no triangles. Removal lemmas play a very important role in extremal and additive combinatorics. As for the of variables, the aspect that is of interest here is that there are many interesting theorems that state that if a function has a certain property, it must be because in a certain sense it depends on only a few variables. For example, a famous result of the first author shows that if a monotone graph property fails to have a sharp threshold, then the property can be approximated by one for which the minimal graphs that satisfy it are all small, with the result that the influence of individual edges on whether the graph has the property is substantial. The main result of this paper is an removal lemma for various families of graphs. It states that if $G$ is a graph in one of these families, then for every set $X$ of vertices in $G$ that spans only a few edges one can throw away just a small proportion of $X$ to obtain a subset $X'$ that is independent. Of course, a result like this is completely false in general: one has only to take a sparse random graph, and then every subset will span few edges but no large subset will be independent. But it is true, for example, for complete bipartite graphs, since a set of vertices that spans a sparse subgraph will have to be contained almost entirely in one of the two vertex sets, and then one can throw away the vertices that are in the other set. One of the families for which the authors prove an edge-removal is that of Kneser graphs. For $0<m<n/2$, the Kneser graph $K(n,m)$ has as its vertex set the set of all subsets of $\{1,2,\dots,n\}$ of size $m$, with two such subsets joined by an edge if and only if they are disjoint. Kneser graphs have large sparse sets, but the natural examples seem to depend on just a few variables: for example, one can take all sets that contain a particular element. (Note that an independent set in $K(n,m)$ is just an intersecting family of $m$-sets.) The authors wanted to prove a precise conjecture that would say that every set of vertices that spans a sparse subgraph can be approximated by a set of vertices that (i) depends on just a few variables and (ii) spans an independent set. Before this paper a weaker statement was known where instead of (ii) one had a sparse set. In this paper the authors settle the conjecture completely by showing that a sparse set of vertices can be approximated by an independent set of vertices. The other family of graphs for which the authors prove an edge removal is that of product graphs. If $H$ is a graph, then the product $G=H^n$ is the graph where $x$ and $y$ are joined if for every coordinate $i$ we have that $x_iy_i$ is an edge of $H$. (There are different definitions of product graphs, but this is the one to which the result applies: note that it makes it quite hard for two vertices to be joined.) For example, if $H$ is a complete graph on $r$ coordinates, then $G=[r]^n$ with two vertices joined if and only if they differ in every coordinate. The proof of the removal uses ideas from Jacob Fox's proof of the triangle removal lemma, which obtains the best known bound for that result. Finally, what has this to do with Swiss cheese? The rough idea is that these graphs have large and highly structured independent sets, and a part of the graph is sparse if and only if it has a big intersection with one of these independent sets. Likewise a Swiss cheese contains lots of large holes, and a region of space contained within the boundary of the cheese will contain only a small amount of cheese if and only if it is almost entirely contained in one of the holes. Article image [by G. J. Wanderer](https://www.flickr.com/photos/106561245@N07/13515457313/in/photolist-mAjfA6-f7mF8i-Tf5Ler-7i4FtY-bi74Nt-gWVXdj-7XoC4N-okRvp5-77HxiZ-7xt4hx-RSUt3B-qeB17D-as2aAc-7K64uq-oL2GMj-6trVsK-UDq452-SP1xSX-d4u57o-fATSUi-8Km2Qs-dJeLzb-qu9Z4Z-dNNvQ2-7eVY4b-6mtCks-frWJEF-aCerNx-7tyjAW-haiMTG-oysrog-9LJVj8-fzno-4VuER1-b5n85M-Jka6K5-a7vHJZ-5S7Yyt-Uwa6kC-dNU7Mm-5AWGod-YSE7a-bzagV7-8tV58z-8YW1XL-5nvECU-QAdC69-8wG84t-TSsHP4-9eoj6S) and distributed under a [CC-BY-NC-SA licence](https://creativecommons.org/licenses/by-nc-sa/2.0/).

Forwarding and optical indices of 4-regular circulant networks