Related papers: A graphon counter example
We construct a sequence of finite graphs that weakly converge to a Cayley graph, but there is no labelling of the edges that would converge to the corresponding Cayley diagram. A similar construction is used to give graph sequences that…
A minimal counterexample to the Erd\H{o}s-Gy\'arf\'as conjecture is a graph of minimum possible order and size with minimum degree at least 3 that contains no cycle whose length is a power of 2. Markstr\"om observed that any such graph must…
We give an example of a non-trivial linear operad that only admits trivial coalgebras and give sufficient conditions ensuring that the cofree coalgebra functor be faithful.
We show that s-convergence of graph sequences is equivalent to the convergence of certain compact sets, called shapes, of Borel probability measures. This result is analogous to the characterization of graphon convergence (with respect to…
We introduce a refutation graph calculus for classical first-order predicate logic, which is an extension of previous ones for binary relations. One reduces logical consequence to establishing that a constructed graph has empty extension,…
Greb\'ik and Rocha [Fractional Isomorphism of Graphons, Combinatorica 42, pp 365-404 (2022)] extended the well studied notion of fractional isomorphism of graphs to graphons. We prove that fractionally isomorphic graphons can be…
Signal analysis on graphs relies heavily on the graph Fourier transform, which is defined as the projection of a signal onto an eigenbasis of the associated shift operator. Large graphs of similar structure may be represented by a graphon.…
Graphons are symmetric measurable functions that arise from a sequence of graphs. A graphon variety is the a set of all graphons defined by a condition of the form $t(g, W) = 0$ for a fixed quantum graph $g$, where $t(.,.)$ is the…
In this paper, we give a simple counter example to the famous Hodge conjecture.
We show an example of a non-symmetric integro-differential equation of order $\alpha$, for $\alpha \in (0,1)$, for which H\"older estimates do not hold even though the kernels are comparable to the fractional Laplacian.
Duffin and Schaeffer provided a famous counterexample to show that Khintchine's theorem fails without monotonicity assumption. Given any monotonically decreasing approximation function with divergent series, we construct…
We give upper and lower bounds on the number of graphs of fixed degree which have a positive density of triangles. In particular, we show that there are very few such graphs, when compared to the number of graphs without this restriction.…
We give an example of a convex, finite and lower semicontinuous function whose subdifferential is everywhere empty. This is possible since the function is defined on an incomplete normed space. The function serves as a universal…
An anagram is a word of the form $WP$ where $W$ is a non-empty word and $P$ is a permutation of $W$. We study anagram-free graph colouring and give bounds on the chromatic number. Alon et al. (2002) asked whether anagram-free chromatic…
Large-scale graph machine learning is challenging as the complexity of learning models scales with the graph size. Subsampling the graph is a viable alternative, but sampling on graphs is nontrivial as graphs are non-Euclidean. Existing…
We characterize the equivalence and the weak equivalence of Cayley graphs for a finite group $\C{A}$. Using these characterizations, we find degree distribution polynomials for weak equivalence of some graphs including 1) circulant graphs…
In [1] the problem of finding a sharp lower bound on lower against number of a general graph is mentioned as an open question. We solve the problem by establishing a tight lower bound on lower against number of a general graph in terms of…
We consider the problem of estimating graph limits, known as graphons, from observations of sequences of sparse finite graphs. In this paper we show a simple method that can shed light on a subset of sparse graphs. The method involves…
We say that a plane set $A$ is {\it graph-null,} if there is a function $g\colon [0,1] \to \mathbb{R}$ such that $\lambda_2 (A+{\rm graph}\, g)=0$. A plane set $A$ has the {\it translational Kakeya property} if, for every translated copy…
We highlight a topological aspect of the graph limit theory. Graphons are limit objects for convergent sequences of dense graphs. We introduce the representation of a graphon on a unique metric space and we relate the dimension of this…