Related papers: On Expanders Graphs: Parameters and Applications
An identifying code of a graph is a subset of its vertices such that every vertex of the graph is uniquely identified by the set of its neighbours within the code. We study the edge-identifying code problem, i.e. the identifying code…
We construct the first explicit two-sided vertex expanders that bypass the spectral barrier. Previously, the strongest known explicit vertex expanders were given by $d$-regular Ramanujan graphs, whose spectral properties imply that every…
Length-constrained expander decompositions are a new graph decomposition that has led to several recent breakthroughs in fast graph algorithms. Roughly, an $(h, s)$-length $\phi$-expander decomposition is a small collection of length…
An orientation of an undirected graph $G$ is an assignment of exactly one direction to each edge of $G$. The oriented diameter of a graph $G$ is the smallest diameter among all the orientations of $G$. The maximum oriented diameter of a…
A common model for social networks are Geometric Inhomogeneous Random Graphs (GIRGs), in which vertices draw a random position in some latent geometric space, and the probability of two vertices forming an edge depends on their geometric…
We consider a game in which a cop searches for a moving robber on a connected graph using distance probes, which is a slight variation on one introduced by Seager. Carragher, Choi, Delcourt, Erickson and West showed that for any $n$-vertex…
For any $\epsilon > 0$, we show that if $G$ is a regular graph on $n \gg_\epsilon 1$ vertices that is $\epsilon$-far (differs by at least $\epsilon n^2$ edges) from any Tur\'{a}n graph, then its second eigenvalue $\lambda_2$ satisfies…
In this paper we present an extension of known semidefinite and linear programming upper bounds for spherical codes and consider a version of this bound for distance graphs. We apply the main result for the distance distribution of a…
A proper vertex coloring of a graph is said to be locally identifying if the sets of colors in the closed neighborhood of any two adjacent non-twin vertices are distinct. The lid-chromatic number of a graph is the minimum number of colors…
The best degree-based upper bound for the spectral radius is due to Liu and Weng. This paper begins by demonstrating that a (forgotten) upper bound for the spectral radius dating from 1983 is equivalent to their much more recent bound. This…
We construct (k+-1)-regular graphs which provide sequences of expanders by adding or substracting appropriate 1-factors from given sequences of k-regular graphs. We compute numerical examples in a few cases for which the given sequences are…
The size of the smallest $k$-regular graph of girth at least $g$ is denoted by the well-studied function $n(k,g)$. We introduce an analogous function $n(H,g)$, defined as the smallest size graph of girth at least $g$ that is a lift (or…
In this note we elaborate on some notions of surface area for discrete graphs which are closely related to the inverse degree. These notions then naturally lead to associated connectivity measures of graphs and to the definition of a…
The segment number of a planar graph $G$ is the smallest number of line segments needed for a planar straight-line drawing of $G$. Dujmovi\'c, Eppstein, Suderman, and Wood [CGTA'07] introduced this measure for the visual complexity of…
We give the first construction of explicit constant-degree lossless vertex expanders. Specifically, for any $\varepsilon > 0$ and sufficiently large $d$, we give an explicit construction of an infinite family of $d$-regular graphs where…
A paradigm that was successfully applied in the study of both pure and algorithmic problems in graph theory can be colloquially summarized as stating that "any graph is close to being the disjoint union of expanders". Our goal in this paper…
Recently, Balla, Janzer, and Sudakov showed a lower bound on the MaxCut in terms of the vector chromatic number, recovering known results on the MaxCut of $H$-free graphs. In this note, we show that their bound is tight, providing a…
In communication field, an important issue is to group users and base stations to as many as possible subnetworks satisfying certain interference constraints. These problems are usually formulated as a graph partition problems which…
We consider a variant of the Cops and Robber game, in which the robber has unbounded speed, i.e. can take any path from her vertex in her turn, but she is not allowed to pass through a vertex occupied by a cop. Let c_{infty}(G) denote the…
Bonnet, Kim, Thomass\'{e}, and Watrigant (2020) introduced the twin-width of a graph. We show that the twin-width of an $n$-vertex graph is less than $(n+\sqrt{n\ln n}+\sqrt{n}+2\ln n)/2$, and the twin-width of an $m$-edge graph for a…